Algebra in Math: Definition, Branches, Basics and Examples

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justify;"><b><strong>Algebra</strong></b><span> is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.</span></p> <p dir="ltr" style="text-align: justify;"><span>It helps represent problems or situations in the form of mathematical expressions which can be solved by using algebraic principles. It is different from Arithmetic as Arithmetic deals with specific numbers and simple operations such as addition and subtraction. Algebra, on the other hand, introduces more complex operations and includes the use of variables, equations, and functions.</span></p><div id="GFG_AD_gfg_mobile_336x280" style="margin: 5px 0;"></div> <p><gfg-carousel><gfg-carousel-content alt="2.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241122163005279969/2.webp" width="800" height="350"></gfg-carousel-content><gfg-carousel-content alt="1.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241122163005124420/1.webp" width="800" height="350"></gfg-carousel-content><gfg-carousel-content alt="Real-LIfe-Applications-of-Algebra-.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241122164708833097/Real-LIfe-Applications-of-Algebra-.webp" width="800" height="350"></gfg-carousel-content></gfg-carousel></p> <blockquote> <p dir="ltr"><b><strong>Example:</strong></b><span> A father is 30 years older than his son. The combined age of both father and son is 60 years. You are asked to find the individual ages of the father and the son.</span></p> <p dir="ltr"><b><strong>Real-life Equation:</strong></b></p> <ul> <li value="1"><span>Let x represent the son’s age.</span></li> <li value="2"><span>The father’s age will then be x + 30, since the father is 30 years older than the son.</span></li> <li value="3"><span>The sum of their ages is 60, so the equation becomes: x + (x + 30) = 60</span></li> </ul> <p dir="ltr"><b><strong>Algebraic Equation:</strong></b><span> 2x + 30 = 60.</span></p> </blockquote> <h2><span>Algebra for Beginners & School Students</span></h2> <ul> <li value="1"><a href="https://www.geeksforgeeks.org/basics-of-algebra/" target="_blank" rel="noopener"><span>Basics of Algebra</span></a></li> <li value="2"><a href="https://www.geeksforgeeks.org/branches-of-algebra/" target="_blank" rel="noopener"><span>Branches of Algebra</span></a></li> <li value="3"><a href="https://www.geeksforgeeks.org/algebraic-expressions/" target="_blank" rel="noopener"><span>Algebraic Expressions</span></a></li> <li value="4"><a href="https://www.geeksforgeeks.org/algebraic-equations/?preview_id=1347345&preview=true" target="_blank" rel="noopener"><span>Algebraic Equations</span></a></li> <li value="5"><a href="https://www.geeksforgeeks.org/algebraic-operations-and-properties/" target="_blank" rel="noopener"><span>Algebraic Operations & Properties</span></a></li> <li value="6"><a href="https://www.geeksforgeeks.org/algebraic-identities/" target="_blank" rel="noopener"><span>Algebraic Identities</span></a></li> <li value="7"><a href="https://www.geeksforgeeks.org/algebra-vs-calculus/" target="_blank" rel="noopener"><span>Algebra vs Calculus</span></a></li> </ul> <h2><span>Algebra for Aptitude Preparation</span></h2> <ul> <li value="1"><a href="https://www.geeksforgeeks.org/algebra-formulas/" target="_blank" rel="noopener"><span>Algebra Formulas</span></a></li> <li value="2"><a href="https://www.geeksforgeeks.org/tips-tricks-for-algebra/" target="_blank" rel="noopener"><span>Algebra Tricks for Aptitude</span></a></li> <li value="3"><a href="https://www.geeksforgeeks.org/algebra-aptitude-questions/" target="_blank" rel="noopener"><span>Algebra Aptitude Questions & Answers</span></a></li> </ul> <h2><span>Algebra Practice Questions</span></h2> <ul> <li value="1"><a href="https://www.geeksforgeeks.org/algebra-practice-questions-easy" target="_blank" rel="noopener"><span>Algebra Practice Questions (Easy Level)</span></a></li> <li value="2"><a href="https://www.geeksforgeeks.org/algebra-practice-questions-medium/" target="_blank" rel="noopener"><span>Algebra Practice Questions (Medium Level)</span></a></li> <li value="3"><a href="https://www.geeksforgeeks.org/algebra-practice-questions-hard-level/" target="_blank" rel="noopener"><span>Algebra Practice Questions (Hard Level)</span></a></li> <li value="4"><a href="https://www.geeksforgeeks.org/quizzes/algebra-gq/" target="_blank" rel="noopener"><span>Algebra Practice Quiz </span></a></li> </ul> <h2><span>Algebra for Programmers</span></h2> <ul> <li value="1"><a href="https://www.geeksforgeeks.org/find-number-of-solutions-of-a-linear-equation-of-n-variables/" rel="noopener"><span>Find number of solutions of a linear equation of n variables</span></a></li> <li value="2"><a href="https://www.geeksforgeeks.org/program-to-find-number-of-solutions-in-quadratic-equation/" rel="noopener"><span>Program to find number of solutions in Quadratic Equation</span></a></li> <li value="3"><a href="https://www.geeksforgeeks.org/program-to-find-the-roots-of-quadratic-equation/" rel="noopener"><span>Program to find the Roots of Quadratic equation</span></a></li> <li value="4"><a href="https://www.geeksforgeeks.org/find-missing-value-from-the-given-equation-a-b-c/" rel="noopener"><span>Find the missing value from the given equation a + b = c</span></a></li> <li value="5"><a href="https://www.geeksforgeeks.org/maximum-minimum-values-algebraic-expression/" rel="noopener"><span>Maximum and Minimum Values of an Algebraic Expression</span></a></li> </ul> <h2><span>Conclusion</span></h2> <p dir="ltr"><span>Algebra is a foundational branch of mathematics that plays a crucial role in solving a wide range of real-world problems. From simple arithmetic to complex systems of equations, algebra provides the tools to model relationships, make predictions, and solve practical challenges. Whether it’s used for budgeting, calculating travel time, or understanding the relationship between ages, algebra is indispensable in daily life, technology, and science. By mastering algebraic concepts, individuals gain problem-solving skills that are essential in fields such as engineering, economics, computer science, and more. </span></p><div id="GFG_AD_Desktop_InContent_ATF_336x280" style="text-align:center; max-height: 280px;"></div><div id="GFG_AD_gfg_mobile_336x280_1" style="margin: 5px 0;"></div> <h2><span>Algebra – FAQs </span></h2> <h3><span>What is Algebra?</span></h3> <blockquote> <p dir="ltr" style="text-align: justify;"><span> Algebra is the area of mathematics that deals with the representation of mathematical statements as solutions to problems. To create a meaningful mathematical expression, it takes variables like x, y, and z together with mathematical operations like addition, subtraction, multiplication, and division.</span></p><div id="_GFG_ABP_Incontent_728x90" style="text-align:center;"></div> </blockquote> <h3><span>What are the Branches of Algebra?</span></h3> <blockquote> <p dir="ltr"><span>The branches of Algebra are:</span></p> <ol> <li value="1"><span>Pre Algebra</span></li> <li value="2"><span>Elementary Algebra</span></li> <li value="3"><span>Abstract Algebra</span></li> <li value="4"><span>Universal Algebra</span></li> <li value="5"><span>Linear Algebra</span></li> <li value="6"><span>Commutative Algebra </span></li> </ol> </blockquote> <h3><span>Why are students required to learn Algebra?</span></h3> <blockquote> <p dir="ltr"><span>Algebra is important for students to study because it develops problem-solving abilities, prepares them for higher-level arithmetic, helps in understanding and analyzing real-world problems, and has applications in a wide range of professions, including science, engineering, economics, and finance. A lot of college majors and jobs also require Algebra.</span></p> </blockquote> <h3><span>Where is algebra used in real life?</span></h3> <blockquote> <p dir="ltr"><span>Algebra is used in everyday life to solve problems involving finances, measurements, and data analysis. It helps with tasks like budgeting, calculating distances, determining interest rates, and managing investments. </span></p> </blockquote> <h3><span>What are the basic operations in Algebra?</span></h3> <blockquote> <p dir="ltr"><span>Basic operations performed in Algebra include addition, subtraction, multiplication, and division of Algebraic equations and solving the equations. </span></p> </blockquote> <h3><span>What is Definition of Algebra? </span></h3> <blockquote> <p dir="ltr"><span>Algebra is a branch of mathematics that deals with symbols and the rules to solve equations and it focuses on operations with variables, constants, and mathematical expressions.</span></p> </blockquote> <h3><span>What is Meaning of Algebra?</span><b><strong> </strong></b></h3> <blockquote> <p dir="ltr"><span>The word Algebra is derived from an Arabic word, ‘Al-jabr’ which means the ‘reunion of broken parts’. So, the meaning of Algebra is </span><b><strong>finding the unknown</strong></b><span>.</span></p> </blockquote> <div class='article_bottom_text'></div><br> <div id="video-tab-content" class="video-tab-content"> <div style="text-align: center; margin: 20px 0px;" id="GFG_AD_InContent_Desktop_728x280"></div> </div> <div class="article-meta-author-details"> <div class="article-meta-author-details-block"> <div class="article-meta-author-details-profile-display"> <div class="article-meta-author-details-profile-display-icon"> <div class="image-wrap"> <img src= https://media.geeksforgeeks.org/wp-content/uploads/20200717172614/authPreLogo.png style="width:35px; height: 35px;border-radius: 25px;" class="avatarprofile_image restrict-popup-gfg" alt="https://media.geeksforgeeks.org/auth/avatar.png"> </div> </div> <div class="article-meta-author-details-profile-display-name" > <div class="info" style="position: unset;"> <div class="name"> <span style="color:#22A779; font-size: 16px; font-weight: 600">GeeksforGeeks</span> </div> </div> </div> </div> </div> <div class="article_bottom_suggestion_wrapper"> <div class="article_bottom_suggestion" data-title="Follow"> <a href='https://news.google.com/publications/CAAqBwgKMLTrzwsw44bnAw?hl=en-IN&gl=IN&ceid=IN%3Aen' target="_blank"> <img class="gfg-icon_Gnews no-zoom-in-cursor" src=https://media.geeksforgeeks.org/auth-dashboard-uploads/Google-news.svg loading="lazy" alt="News"></img> </a> </div> <div class='discussion_panel' data-title='Comments'> <i class='discussion_button' data-gfg-action='loadComments'> </i> </div> <div pid="612557" class="article--viewer_like tooltip tooltipBottom" data-title="Like Article">  <button id="likeButton" aria-label="like article" data-gfg-action="like-article" data-bookmark-value="0" data-liked="false" style="color: #5B5A5A !important; display: flex; align-items: center; background-color: unset; margin: 0px !important; padding: 5px 0px;"> <i class="author-badge-like-button"></i> <figure id="likeCount" style="margin-left: 3px; margin-top: 4px; color: var(--like-count-color); font-size: 14px; font-weight: 600;" class="favoriteText"></figure> </button> </div> <div class="article_bottom_suggestion" onclick="improveArticleCall()" data-title="Improve"> <span class="improveTooltipBottom">Improve</span> <i class="author-badge-improvement-button"></i> </div> </div> </div> <div class="d-row content-bw "style="margin-top: 20px;"> <div class="article-pgnavi_prev"> </div> <div class="article-pgnavi_next"> <a href="https://www.geeksforgeeks.org/what-is-a-variable-in-algebra/?ref=next_article" class="pg-head"> <span style="margin-right: 5px; margin-left: auto;">Next Article</span> <span class="gfg-icon gfg-icon_next"></span> </a>  <div class="pg-main"> <a href="https://www.geeksforgeeks.org/what-is-a-variable-in-algebra/?ref=next_article">What is a variable in algebra?</a> </div> </div> </div> </div> </article> <div class="article--recommended article--recommended_wrapper" id="similar-reads"> <h3 class="new-top-bar top-bar-title">Similar Reads</h3> <div class="gfg-similar-reads-list"> <ul class="similarReadLeftBarList"><div class='second'><li class="currentpage similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebra in Math: Definition, Branches, Basics and Examples </div> <div class="gfg-similar-read-item-subheading">Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. 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The most general form of a polynomial is given</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/division-algorithm-for-polynomials/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Division Algorithm for Polynomials </div> <div class="gfg-similar-read-item-subheading">Polynomials are those algebraic expressions that contain variables, coefficients, and constants. For Instance, in the polynomial 8x2 + 3z - 7, in this polynomial, 8,3 are the coefficients, x and z are the variables, and 7 is the constant. Just as simple Mathematical operations are applied on numbers</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/division-algorithm-problems-and-solutions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Division Algorithm Problems and Solutions </div> <div class="gfg-similar-read-item-subheading">Polynomials are made up of algebraic expressions with different degrees. Degree-one polynomials are called linear polynomials, degree-two are called quadratic and degree-three are called cubic polynomials. Zeros of these polynomials are the points where these polynomials become zero. 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Instead of performing long or synthetic division, you can use this theorem to substitute the polynomial and get the remainder d</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/factor-theorem/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Factor Theorem </div> <div class="gfg-similar-read-item-subheading">Factor theorem is used for finding the roots of the given polynomial. This theorem is very helpful in finding the factors of the polynomial equation without actually solving them. According to the factor theorem, for any polynomial f(x) of degree n ≥ 1 a linear polynomial (x - a) is the factor of th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebraic-identities-of-polynomials/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebraic Identities of Polynomials </div> <div class="gfg-similar-read-item-subheading">Algebraic identities are equations that hold true for all values of the variables involved. In the context of polynomials, these identities are particularly useful for simplifying expressions and solving equations. What are Algebraic Identities?Algebraic Identities are defined for the algebraic expr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/factoring-polynomials/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Factoring Polynomials </div> <div class="gfg-similar-read-item-subheading">Factoring Polynomials: A basic algebraic concept called factoring polynomials involves breaking down a polynomial equation into simpler parts. Factoring can be used to solve equations, simplify complicated expressions, and locate the roots or zeros of polynomial functions. In several fields of math</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/relationship-between-zeroes-and-coefficients-of-a-polynomial/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Relationship between Zeroes and Coefficients of a Polynomial </div> <div class="gfg-similar-read-item-subheading">Polynomials are algebraic expressions with constants and variables that can be linear i.e. the highest power o the variable is one, quadratic and others. The zeros of the polynomials are the values of the variable (say x) that on substituting in the polynomial give the answer as zero. While the coef</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Linear Equations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/linear-equations-in-one-variable/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Linear Equations in One Variable </div> <div class="gfg-similar-read-item-subheading">Linear equation in one variable is the equation that is used in algebra for finding unknown quantities. It is used for representing the conditions that are dependent on one variable. It is a linear equation i.e. the equation in which the degree of the equation is one, and it only has one variable. L</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/solve-linear-equations-with-variable-on-both-sides/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Solve Linear Equations with Variable on both Sides </div> <div class="gfg-similar-read-item-subheading">Equations contain two types of quantities, one variable the other is number. But we can consider both equivalents for the sake of reverse operations. This is one of the important things that a student must keep in mind for solving various problems. Also, a good understanding of reverse operations wo</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/graphical-methods-of-solving-pair-of-linear-equations-in-two-variables/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Graphical Methods of Solving Pair of Linear Equations in Two Variables </div> <div class="gfg-similar-read-item-subheading">A system of linear equations is just a pair of two lines that may or may not intersect. The graph of a linear equation is a line. There are various methods that can be used to solve two linear equations, for example, Substitution Method, Elimination Method, etc. An easy-to-understand and beginner-fr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/reducing-equations-to-simpler-form-class-8-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Reducing Equations to Simpler Form | Class 8 Maths </div> <div class="gfg-similar-read-item-subheading">Reducing equations is a method used to simplify complex equations into a more manageable form. This technique is particularly useful when dealing with non-linear equations, which cannot always be solved directly. By applying specific mathematical operations, such as cross-multiplication, these equat</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/linear-equation-in-two-variables/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Linear Equation in Two Variables </div> <div class="gfg-similar-read-item-subheading">Linear Equation in Two Variables: A Linear equation is defined as an equation with the maximum degree of one only, for example, ax = b can be referred to as a linear equation, and when a Linear equation in two variables comes into the picture, it means that the entire equation has 2 variables presen</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/graph-of-linear-equations-in-two-variables/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Graph of Linear Equations in Two Variables </div> <div class="gfg-similar-read-item-subheading">Linear equations are the first-order equations, i.e. the equations of degree 1. The equations which are used to define any straight line are linear, they are represented as, x + k = 0; These equations have a unique solution and can be represented on number lines very easily. Let's look at linear e</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/equations-of-lines-parallel-to-the-x-axis-and-y-axis/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Equations of Lines Parallel to the x-axis and y-axis </div> <div class="gfg-similar-read-item-subheading">Linear Equations allow us to explain a lot of physical phenomena happening around us. For example, A train running between two stations at a constant speed, the speed of a falling object. Even the straight lines we draw on paper can be represented in form of linear equations mathematically. A linear</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/pair-of-linear-equations-in-two-variables/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Pair of Linear Equations in Two Variables </div> <div class="gfg-similar-read-item-subheading">Linear Equation in two variables are equations with only two variables and the exponent of the variable is 1. This system of equations can have a unique solution, no solution, or an infinite solution according to the given initial condition. Linear equations are used to describe a relationship betwe</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/number-of-solutions-to-a-system-of-equations-algebraically/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Number of Solutions to a System of Equations Algebraically </div> <div class="gfg-similar-read-item-subheading">A statement that two mathematical expressions of one or more variables are identical is called an equation. Linear equations are those in which the powers of all the variables concerned are equal. A linear equation's degree is always one. A solution of the simultaneous pair of linear equations is a</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/substitution-method/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Solve the Linear Equation using Substitution Method </div> <div class="gfg-similar-read-item-subheading">Solving Linear Equation using the Substitution Method: The equation in which the highest power of the variable is always 1 is called a linear equation (or) the first-order equation. A linear equation's graph will always be a straight line. When the equation has only one variable and the highest degr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/cross-multiplication-method/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Cross Multiplication Method </div> <div class="gfg-similar-read-item-subheading">Cross multiplication method is one of the basic methods in mathematics that is used to solve the linear equations in two variables. It is one of the easiest to solve a pair of linear equations in two variables. Suppose we have a pair of linear equations in two variables, i.e. a1x + b1y = -c1 and a2</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/equations-reducible-to-linear-form/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Equations Reducible to Linear Form </div> <div class="gfg-similar-read-item-subheading">Equations Reducible to Linear Form" refers to equations that can be transformed or rewritten into a linear equation. These equations typically involve variables raised to powers other than 1, such as squared terms, cubed terms, or higher. By applying suitable substitutions or transformations, these</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Quadratic Equations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/quadratic-equation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Quadratic Equations </div> <div class="gfg-similar-read-item-subheading">Quadratic equation is a fundamental concept of algebra and mathematics, It is a second-degree equation that can be represented as ax2 + bx + c = 0. In this equation, x is an unknown variable, a, b, and c are constants, and a is not equal to 0. A quadratic Equation can be solved by using methods such</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/solving-quadratic-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Solving Quadratic Equations </div> <div class="gfg-similar-read-item-subheading">Solving quadratic equations is a fundamental skill in mathematics that plays a crucial role in various fields such as physics, engineering, and finance. As Quadratic Equations occur almost everywhere in our real life. For example, even the problem of designing a playground can be formulated as a qua</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/roots-of-quadratic-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Roots of Quadratic Equation </div> <div class="gfg-similar-read-item-subheading">Roots of Quadratic Equations are the values of the variable for which the quadratic equation gets satisfied. The roots of a quadratic equation are also called zeros of a quadratic equation. A quadratic equation is an equation in which the maximum power of a variable is 2. Hence, there can be a maxim</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Cubic Equations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/cubic-equation-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Cubic Equation Formula </div> <div class="gfg-similar-read-item-subheading">A cubic equation is a polynomial equation of degree three, and it can be written in the general form: ax3 + bx2 + cx + d = 0. Solutions to a cubic equation can be found using various methods, including factoring, synthetic division, or using the cubic formula. In this article, we will learn about cu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/solving-cubic-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Solving Cubic Equations </div> <div class="gfg-similar-read-item-subheading">Cubic Equation is a mathematical equation in which a polynomial of degree 3 is equated to a constant or another polynomial of maximum degree 2. The standard representation of the cubic equation is ax3+bx2+cx+d = 0 where a, b, c, and d are real numbers. Some examples of cubic equation are x3 - 4x2 +</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Sequence and Series<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-arithmetic-progression/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Arithmetic Progression </div> <div class="gfg-similar-read-item-subheading">Arithmetic progression (AP) is simply a sequence of numbers such that the difference between any two consecutive terms is constant. Real World Examples of AP Natural Numbers: 1, 2, 3, 4, 5, . . . with a common difference 1Even Numbers: 2, 4, 6, 8, 10, . . . with a common difference 2Odd Numbers: 1,</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/arithmetic-series/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Arithmetic Series </div> <div class="gfg-similar-read-item-subheading">An arithmetic series is the sum of the terms of an arithmetic sequence, where an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/arithmetic-sequence/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Arithmetic Sequence </div> <div class="gfg-similar-read-item-subheading">In the arithmetic sequence, the absolute difference between one term and the next term is constant. In other words, a sequence a1, a2, . . ., an is called an arithmetic sequence or arithmetic progression if an+1 - an = d where d is constant and it is the common difference of the sequence. Let's make</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/geometric-progression/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Program to Check Geometric Progression </div> <div class="gfg-similar-read-item-subheading">A sequence of numbers is called a Geometric progression if the ratio of any two consecutive terms is always the same. In simple terms, A geometric series is a list of numbers where each number, or term, is found by multiplying the previous term by a common ratio r. The general form of Geometric Prog</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/geometric-series/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Geometric Series </div> <div class="gfg-similar-read-item-subheading">In a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease. Geometric Sequence is given as: a, ar, ar2, ar3, ar4,..... {Infinite Sequence}a, ar, ar2, ar3, ar4, ....... arn</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Set Theory<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/representation-of-a-set/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Representation of a Set </div> <div class="gfg-similar-read-item-subheading">Imagine a very haphazard world where no categories are divided to memorize and classify things separately, a world like this will be full of chaos and mess, this is why humans prefer to categorize things and classify them to neatly understand and remember them. The same case happens in mathematics,</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/types-of-sets/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Types Of Sets </div> <div class="gfg-similar-read-item-subheading">Sets are a well-defined collection of objects. Objects that a set contains are called the elements of the set. We can also consider sets as collections of elements that have a common feature. For example, the collection of even numbers is called the set of even numbers. Table of Content What is Set?</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/universal-set/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Universal Sets </div> <div class="gfg-similar-read-item-subheading">Universal Set is a set that has all the elements associated with a given set, without any repetition. Suppose we have two sets P = {1, 3, 5} and Q = {2, 4, 6} then the universal set of P and Q is U = {1, 2, 3, 4, 5, 6}. We generally use U to denote universal sets. Universal Set is a type of set that</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/venn-diagram/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Venn Diagram </div> <div class="gfg-similar-read-item-subheading">Venn Diagrams are used for the visual representation of relationships as they provide a clear, visual method for showing how different sets intersect, overlap, or remain distinct. They are essential tools in mathematics and logic for illustrating the relationships between sets. By employing intersec</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/operations-on-sets/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Operations on Sets </div> <div class="gfg-similar-read-item-subheading">Sets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this article, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/union-of-sets/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Union of Sets </div> <div class="gfg-similar-read-item-subheading">Union of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union. Example 1:If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,A∪B = {1, 2, 3, 5, 7} Example 2:If A = {1, 3. 5.7} and B = {2, 4}</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/cartesian-product-of-sets/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Cartesian Product of Sets </div> <div class="gfg-similar-read-item-subheading">'Product' mathematically signifies the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5. One must be familiar with the basic operations on sets like Union and Intersection, which are performed on 2 or more sets. Cartesian Product is also one</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Relations and Functions<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/relations-and-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Relation and Function </div> <div class="gfg-similar-read-item-subheading">Relation and Function are two ways of establishing links between two sets in mathematics. Relation and Function in maths are analogous to the relation that we see in our daily lives i.e., two persons are related by the relation of father-son, mother-daughter, brother-sister, and many more. On a simi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-a-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">What is a Function in Maths? </div> <div class="gfg-similar-read-item-subheading">A function in maths is a rule that relates one value, called the input, to another value, called the output. In simple terms, it’s like a machine that takes something in, processes it, and gives something out. Every input has a specific output. Functions are a fundamental part of mathematics and are</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/types-of-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Types of Functions </div> <div class="gfg-similar-read-item-subheading">Functions are defined as the relations which give a particular output for a particular input value. A function has a domain and codomain (range). f(x) usually denotes a function where x is the input of the function. In general, a function is written as y = f(x). Table of Content What is a Function?T</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/composite-functions-relations-and-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Composite functions - Relations and functions </div> <div class="gfg-similar-read-item-subheading">Let f : A->B and g : B->C be two functions. Then the composition of f and g, denoted by g o f, is defined as the function g o f : A->C given by g o f (x) = g{f(x)}, ∀ x ∈ A. Clearly, dom(g o f) = dom(f). Also, g o f is defined only when range(f) is a subset of dom(g). Evaluating composite f</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/invertible-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Invertible Functions </div> <div class="gfg-similar-read-item-subheading">As the name suggests Invertible means "inverse", and Invertible function means the inverse of the function. Invertible functions, in the most general sense, are functions that "reverse" each other. For example, if f takes a to b, then the inverse, f-1, must take b to a. Table of Content Invertible F</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/composition-of-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Composition of Functions </div> <div class="gfg-similar-read-item-subheading">Composition of Function is the process or operation which combines two or more functions together into a single function. We define functions as the set of operations that operate on a set of values and give the desired output. Suppose f is a function say a function which draws the sketch of the fru</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse Functions | Definition, Condition for Inverse and Examples </div> <div class="gfg-similar-read-item-subheading">Inverse Functions are an important concept in mathematics. An inverse function basically reverses the effect of the original function. If you apply a function to a number and then apply its inverse, you get back the original number. For example, if a function turns 2 into 5, the inverse function wil</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/verifying-inverse-functions-by-composition/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Verifying Inverse Functions by Composition </div> <div class="gfg-similar-read-item-subheading">A function can be seen as a mathematical formula or a machine that throws output when an input is given. The output is usually some processed version of the input. Function's inverses can be seen as the operations which give us the input back on giving them the output. In other words, inverse functi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/domain-and-range-of-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Domain and Range of a Function </div> <div class="gfg-similar-read-item-subheading">Domain and Range of a Function: Domain and Range are the input and output values of a Function. A function is defined as the relation between a set of inputs and their outputs, where the input can have only one output i.e. a domain can yield a particular range. It depicts a relationship between an i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/piecewise-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Piecewise Function </div> <div class="gfg-similar-read-item-subheading">Piecewise Function is a function that behaves differently for two types of input. As we know a function is a mathematical object which associates each input with exactly one output. For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as f(x)</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Functions<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/linear-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Linear Function </div> <div class="gfg-similar-read-item-subheading">A linear function is a mathematical function that creates a straight line when graphed. It can be described by the formula: y = mx+b. A linear function in Algebra represents a straight line in the 2-D or 3-D cartesian plane. Hence this function is called a linear function. It is a function with vari</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/exponential-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Exponential Functions: Definition, Formula and Examples </div> <div class="gfg-similar-read-item-subheading">Exponential Function is a mathematical function that involves exponents. It is written in the form f (x) = ax, where “x” is a variable and “a” is a constant. The constant a is the base of the function and it should be greater than 0. An exponential function is classified into two types, i.e., expone</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-are-the-six-trigonometry-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Six Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Trigonometry can be defined as the branch of mathematics that determines and studies the relationships between the sides of a triangle and the angles subtended by them. Trigonometry is used in the case of right-angled triangles. Trigonometric functions define the relationships between the 3 sides an</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse Trigonometric Functions | Definition, Formula, Types and Examples </div> <div class="gfg-similar-read-item-subheading">Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/increasing-and-decreasing-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Increasing and Decreasing Functions </div> <div class="gfg-similar-read-item-subheading">If you're studying calculus, then you're probably familiar with the concepts of increasing and decreasing functions. These terms refer to the behavior of a function as its input values change. Specifically, an increasing function is one that becomes larger as its input values increase, while a decre</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Matrices and Determinants<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/types-of-matrices/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Types of Matrices </div> <div class="gfg-similar-read-item-subheading">Types of Matrices classify matrices into different categories based on the number of rows and columns present in them, the position of the elements, and also the special properties exhibited by the Matrix. A matrix is a rectangular array of numbers in which elements are arranged in rows and columns.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/matrix-operations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Matrix Operations </div> <div class="gfg-similar-read-item-subheading">Matrix Operations are the operations that are operated on the matrix. Matrix Operation includes operations such as Addition of Matrix, Subtraction of Matrix, Multiplication of Matrix, etc, and others. These operations are very b useful for solving various problems of matrices and help us to find the</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/matrix-addition/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Matrix Addition </div> <div class="gfg-similar-read-item-subheading">Matrix Addition as the name suggests in the article, explores the addition of matrices and it is one of the fundamental operations in the field of Linear Algebra. Matrix Addition is very useful in various fields such as data analysis, computer graphics, image processing, cryptography, operations res</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/matrix-multiplication/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Matrix Multiplication </div> <div class="gfg-similar-read-item-subheading">Matrix multiplication is a fundamental operation in mathematics that involves multiplying two or more matrices by following specific rules. Understanding how to multiply matrices is crucial for solving various mathematical problems. In this article, we will learn how to do matrix multiplication, rul</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/transpose-of-a-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Transpose of a Matrix </div> <div class="gfg-similar-read-item-subheading">Transpose of a matrix is a very common method used for matrix transformation in linear algebra. Transpose of a matrix is obtained by interchanging the rows and columns of the given matrix or vice versa. Transpose of a matrix can be utilized to obtain the adjoint and inverse of the matrices. Before l</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-symmetric-matrix-and-skew-symmetric-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Symmetric and Skew Symmetric Matrices </div> <div class="gfg-similar-read-item-subheading">Symmetric and Skew Symmetric Matrices are the types of square matrices based on the relation between a matrix and its transpose. These matrices are one of the most used matrices out of all the matrices out there. Symmetric matrices have use cases in optimization, physics, and statistics, whereas ske</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/elementary-operations-on-matrices/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Elementary Operations on Matrices </div> <div class="gfg-similar-read-item-subheading">Elementary Operations on Matrices are the operations performed on the rows and columns of the matrix that do not change the value of the matrix. A matrix is a way of representing numbers in the form of an array, i.e. the numbers are arranged in the form of rows and columns. In a matrix, the rows and</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inverse-of-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inverse of a Matrix </div> <div class="gfg-similar-read-item-subheading">The inverse of a Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any square matrix A, its inverse is denoted as A-1. The inverse of a matrix is obtained by dividing the adjugate of the given matrix by the determinant of the given matrix. Where I i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/invertible-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Invertible Matrix </div> <div class="gfg-similar-read-item-subheading">Invertible matrices are defined as the matrix whose inverse exists. We define a matrix as the arrangement of data in rows and columns, if any matrix has m rows and n columns then the order of the matrix is m × n where m and n represent the number of rows and columns respectively. We define invertib</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/determinant-of-a-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Determinant of a Matrix </div> <div class="gfg-similar-read-item-subheading">The determinant of a Matrix is defined as a special number that is defined only for square matrices (matrices that have the same number of rows and columns). A determinant is used in many places in calculus and other matrices related to algebra, it actually represents the matrix in terms of a real n</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-determinants-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Determinants </div> <div class="gfg-similar-read-item-subheading">Properties of Determinants are the properties that are required to solve various problems in Matrices. There are various properties of the determinant that are based on the elements, rows, and columns of the determinant. These properties help us to easily find the value of the determinant. Suppose w</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-of-triangle-using-determinants/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area of a Triangle using Determinants </div> <div class="gfg-similar-read-item-subheading">Area of a Triangle using Determinants is one of the multiple methods available to find the area of a triangle and this method is often overlooked for its complexity. But the Area of a Triangle using Determinants comes very handy in coordinate geometry when all three vertices of a triangle are given</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/minors-and-cofactors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Minors and Cofactors </div> <div class="gfg-similar-read-item-subheading">Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. A cofactor is (-1)i+j ti</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/adjoint-of-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Adjoint of a Matrix </div> <div class="gfg-similar-read-item-subheading">Knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. From matrices there come Determinants, Now we see one of the properties of the Determinant in this article. In this article, we see how to find the Adjoint of a Matrix.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/applications-of-matrices-and-determinants/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Applications of Matrices and Determinants </div> <div class="gfg-similar-read-item-subheading">Applications of Matrices and Determinants: One application of matrices and determinants is that they can be used to solve linear equations in two or three variables. Matrices and determinants are also used to check the consistency of any system, whether they are consistent or not. This is the most u</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Vectors<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/vector-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Vector Algebra </div> <div class="gfg-similar-read-item-subheading">Vectors algebra is the branch of algebra that involves operations on vectors. Vectors are quantities that have both magnitude and direction so normal operations are not performed on the vectors. We can add, subtract, and multiply vector quantities using special vector algebra rules. Vectors can be e</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/vector-addition/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Vector Addition </div> <div class="gfg-similar-read-item-subheading">Vector Addition in Mathematics is the fundamental operation of vector algebra that is used to find the sum of two vectors. Vectors are mathematical quantities that have magnitude and direction. A vector can be represented by a line with an arrow pointing towards its direction and its length represen</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/vector-operations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Vector Operations </div> <div class="gfg-similar-read-item-subheading">Vector Operations are operations that are performed on vector quantities. Vector quantities are the quantities that have both magnitude and direction. So performing mathematical operations on them directly is not possible. So we have special operations that work only with vector quantities and hence</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/scalar-product-of-vectors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Scalar Product of Vectors </div> <div class="gfg-similar-read-item-subheading">Two vectors or a vector and a scalar can be multiplied. There are mainly two kinds of products of vectors in physics, scalar multiplication of vectors and Vector Product (Cross Product) of two vectors. The result of the scalar product of two vectors is a number (a scalar). The common use of the scal</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/dot-and-cross-products-on-vectors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Dot and Cross Products on Vectors </div> <div class="gfg-similar-read-item-subheading">A quantity that is characterized not only by magnitude but also by its direction, is called a vector. Velocity, force, acceleration, momentum, etc. are vectors. Vectors can be multiplied in two ways: Scalar product or Dot productVector Product or Cross productTable of Content Scalar Product/Dot Pr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/how-to-find-the-angle-between-two-vectors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">How to Find the Angle Between Two Vectors? </div> <div class="gfg-similar-read-item-subheading">Vector quantities are the physical quantities that have both magnitude and direction and the angle between two vectors can be easily found if the dot product or the cross product of the two vectors is given. In this article, we will learn how to find the angle between two vectors, its formula, relat</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/magnitude-of-a-vector/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Magnitude of a Vector </div> <div class="gfg-similar-read-item-subheading">Vector quantities are the quantities that have both direction and magnitude. The magnitude of a vector is the length of the vector. It is given by the numeric value of the vector and as it represents the length of the vector so it is always positive. For any vector [Tex]\overrightarrow{A} [/Tex] it</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/direction-cosines-and-direction-ratios/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Direction Cosines and Direction Ratios </div> <div class="gfg-similar-read-item-subheading">Usually, for three-dimensional geometry, we rely on the three-dimensional Cartesian plane. Vectors can also be used to describe the lines and the angles they make with the axis. How should we describe a line passing through the origin making an angle with different axes? We define them using cosine</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/resultant-vector-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Resultant Vector Formula: Definition, Examples </div> <div class="gfg-similar-read-item-subheading">Resultant vector formula gives the resultant value of two or more vectors. The result is obtained by computing the vectors with consideration of the direction of each vector to others. This formula has various applications in Engineering & Physics. What is Resultant Vector Formula?A resultant ve</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/vector-projection-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Vector Projection - Formula, Derivation & Examples </div> <div class="gfg-similar-read-item-subheading">Vector Projection is the shadow of a vector over another vector. It allows you to determine how one vector influences another in a specific direction. The projection vector is obtained by multiplying the vector with the Cos of the angle between the two vectors. A vector has both magnitude and direct</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/orthogonal-and-orthonormal-vectors-in-linear-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Orthogonal and Orthonormal Vectors in Linear Algebra </div> <div class="gfg-similar-read-item-subheading">Orthogonal Vectors: Two vectors are orthogonal to each other when their dot product is 0. How do we define the dot product? Dot product(scalar product) of two n-dimensional vectors A and B, is given by this expression.[Tex]A . B=\sum_{i=1}^{n} a_{i} b_{i} [/Tex]Thus the vectors A and B are orthogona</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/distance-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Distance Formula </div> <div class="gfg-similar-read-item-subheading">Distance Formula is a point that is used to find the distance between two points, a point, a line, and two line segments. The distance formula is based on the Pythagorean theorem. the distance formula for the same is: d = √[(x2 - x1)2 + (y2 - y1)2] In this article, we will learn about the dista</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/how-to-find-the-angle-between-two-vectors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">How to Find the Angle Between Two Vectors? </div> <div class="gfg-similar-read-item-subheading">Vector quantities are the physical quantities that have both magnitude and direction and the angle between two vectors can be easily found if the dot product or the cross product of the two vectors is given. In this article, we will learn how to find the angle between two vectors, its formula, relat</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/magnitude-of-a-vector/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Magnitude of a Vector </div> <div class="gfg-similar-read-item-subheading">Vector quantities are the quantities that have both direction and magnitude. The magnitude of a vector is the length of the vector. It is given by the numeric value of the vector and as it represents the length of the vector so it is always positive. For any vector [Tex]\overrightarrow{A} [/Tex] it</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/direction-cosines-and-direction-ratios/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Direction Cosines and Direction Ratios </div> <div class="gfg-similar-read-item-subheading">Usually, for three-dimensional geometry, we rely on the three-dimensional Cartesian plane. Vectors can also be used to describe the lines and the angles they make with the axis. How should we describe a line passing through the origin making an angle with different axes? We define them using cosine</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/direction-of-a-vector-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Direction of a Vector Formula </div> <div class="gfg-similar-read-item-subheading">A vector is formed when two distinct points are joined with each other. The point from which a line was drawn to the other point determines the direction of that vector. A vector's direction is the angle formed by the vector with the horizontal axis, also known as the X-axis. It is provided by the c</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/vector-projection-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Vector Projection - Formula, Derivation & Examples </div> <div class="gfg-similar-read-item-subheading">Vector Projection is the shadow of a vector over another vector. It allows you to determine how one vector influences another in a specific direction. The projection vector is obtained by multiplying the vector with the Cos of the angle between the two vectors. A vector has both magnitude and direct</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/distance-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Distance Formula </div> <div class="gfg-similar-read-item-subheading">Distance Formula is a point that is used to find the distance between two points, a point, a line, and two line segments. The distance formula is based on the Pythagorean theorem. the distance formula for the same is: d = √[(x2 - x1)2 + (y2 - y1)2] In this article, we will learn about the dista</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/section-formula-vector-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Section Formula - Vector Algebra </div> <div class="gfg-similar-read-item-subheading">The section formula in vector algebra is a useful formula for finding a point that divides a line segment into a certain ratio. This concept is particularly important in physics and engineering for tasks such as finding center of mass, centroids, and other applications where a point must be located</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/angle-between-two-vectors-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Angle between Two Vectors Formula </div> <div class="gfg-similar-read-item-subheading">Angle between two vectors is the angle between their tails and this angle can be easily found using cross product and dot product of vector formulas. Angle between two vectors always lies between 0° and 180°. In this article we will learn about, angle between two vectors, definition, formulas, and e</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/program-to-find-equation-of-a-plane-passing-through-3-points/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Program to find equation of a plane passing through 3 points </div> <div class="gfg-similar-read-item-subheading">Given three points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3). The task is to find the equation of the plane passing through these 3 points. Examples: Input: x1 = -1 y1 = w z1 = 1 x2 = 0 y2 = -3 z2 = 2 x3 = 1 y3 = 1 z3 = -4 Output: equation of plane is 26 x + 7 y + 9 z + 3 = 0.Input: x1 = 2, y1 = 1, z</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Plane<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/cartesian-plane/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Cartesian Plane </div> <div class="gfg-similar-read-item-subheading">Cartesian plane is defined as the two-dimensional plane used in the Cartesian coordinate system. This plane is formed by intersecting two perpendicular lines called the x-axis and the y-axis and their intersection is called the origin. This method of distributing the 2-Dimensional space into four ar</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-a-plane-in-geometry/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">What is a plane in geometry? </div> <div class="gfg-similar-read-item-subheading">Geometry is about the observation of shapes, sizes, angles, and dimensions of two-dimensional shapes. These two dimensions of geometrical figures include length and width. The 2D geometries, it is categorized into three types are euclidean geometry, spherical geometry, and hyperbolic geometry. Geome</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">2 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Boolean Algebra<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/boolean-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Boolean Algebra </div> <div class="gfg-similar-read-item-subheading">Boolean algebra is a type of algebra that is created by operating the binary system. In the year 1854, George Boole, an English mathematician, proposed this algebra. This is a variant of Aristotle's propositional logic that uses the symbols 0 and 1, or True and False. Boolean algebra is concerned wi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-boolean-algebra/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Boolean Algebra </div> <div class="gfg-similar-read-item-subheading">The Boolean Algebra uses sets of rules for analyzing digital gates and circuits. In this article, we will be going through the Properties or Laws of the Boolean algebra. So first we will start our article by defining what are the properties of Boolean Algebra, and then we will go through what are Bo</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/boolean-algebraic-theorems/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Boolean Algebraic Theorems </div> <div class="gfg-similar-read-item-subheading">Boolean algebra is a fundamental area of mathematics that deals with variables and logical operations. 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Sometimes these th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div></div></ul> </div> </div> <div class="bottom-wrap" style="padding: 0px 20px;margin-bottom: 10px;"> <div class="improved"> <div class="t-head">Article Tags : </div> <ul> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/school-learning/class-10/?ref=article_category">Class 10</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/school-learning/maths/?ref=article_category">Mathematics</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/school-learning/?ref=article_category">School Learning</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/tag/algebra/?ref=article_tag">Algebra</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/maths-categories/?ref=article_tag">Maths-Categories</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/maths-class-10/?ref=article_tag">Maths-Class-10</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/maths-class-11/?ref=article_tag">Maths-Class-11</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/maths-class-12/?ref=article_tag">Maths-Class-12</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/maths-class-8/?ref=article_tag">Maths-Class-8</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/maths-class-9/?ref=article_tag">Maths-Class-9</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/tutorials/?ref=article_tag">Tutorials</a> </li> </ul> <button id="showMoreTagsBtn" class="show-more-recommendation-button">+7 More</button> </div> </div> <div class="vote-wrap"> <div style="display:none;align-items:center;justify-content:center;width:100%;"> <button aria-label="like" data-type="like" class="vote-this" style="margin-right: 0; 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