{"id":126469,"date":"2022-10-14T18:56:32","date_gmt":"2022-10-14T13:26:32","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=126469"},"modified":"2024-12-13T17:52:16","modified_gmt":"2024-12-13T12:22:16","slug":"dot-product-of-two-vectors-properties-and-examples-2","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/","title":{"rendered":"Dot Product of Two Vectors | Properties and Examples"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Scalar_Dot_Product_of_Two_Vectors\" title=\"Scalar Dot Product of Two Vectors\">Scalar Dot Product of Two Vectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Dot_Product_of_Two_Vectors_-_Explained\" title=\"Dot Product of Two Vectors &#8211; Explained\">Dot Product of Two Vectors &#8211; Explained<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Dot_Product_Definition\" title=\"Dot Product Definition\">Dot Product Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Dot_Product_Formula\" title=\"Dot Product Formula\">Dot Product Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Dot_Product_Geometry_Definition\" title=\"Dot Product Geometry Definition\">Dot Product Geometry Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Dot_Product_Algebra_Definition\" title=\"Dot Product Algebra Definition\">Dot Product Algebra Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Properties_of_Dot_Product_of_Two_Vectors\" title=\"Properties of Dot Product of Two Vectors\">Properties of Dot Product of Two Vectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Dot_Product_of_Vector-Valued_Functions\" title=\"Dot Product of Vector-Valued Functions\">Dot Product of Vector-Valued Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#Solved_Examples\" title=\"Solved Examples\">Solved Examples<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/study-materials\/maths\/scalar-dot-product-of-two-vectors\/#NCERT_Study_Material\" title=\"NCERT Study Material\">NCERT Study Material<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Scalar_Dot_Product_of_Two_Vectors\"><\/span>Scalar Dot Product of Two Vectors<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Scalar Dot Product Of Two Vectors: Dot Product of Two Vectors &#8211; In order to understand the Dot product of two vectors, we need to first understand what a projection is. A projection of one vector on the other is often referred to as the orthogonal projection of one vector on the other.<\/p>\n<p>The <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/dot-product-of-two-vectors\/\">dot product<\/a> of two vectors is a scalar quantity that is calculated by multiplying the magnitude of one vector by the magnitude of the other vector and then by the cosine of the angle between them. The dot product is a measure of the similarity of two vectors and is use in physics to calculate the force between two objects.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-126466 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/Scalar-Dot-Product-Of-Two-Vectors-300x212.jpg\" alt=\"Dot Product of Two Vectors | Properties and Examples\" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/Scalar-Dot-Product-Of-Two-Vectors-300x212.jpg?v=1645815275 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/02\/Scalar-Dot-Product-Of-Two-Vectors.jpg?v=1645815275 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dot_Product_of_Two_Vectors_-_Explained\"><\/span>Dot Product of Two Vectors &#8211; Explained<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The dot product of two vectors is a scalar value that compute by multiplying the components of each vector and then summing the products. The result is a number that represents the magnitude of the vector product and the direction is given by the angle between the vectors.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dot_Product_Definition\"><\/span>Dot Product Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A dot product is a mathematical operation that takes two vectors and multiplies them together to produce a single scalar value. The dot product denote by the symbol . The two vectors place side by side, with the first vector on the left and the second vector on the right. The dot product calculate by multiplying each component of the first vector by the corresponding component of the second vector and then adding all of the resulting products together.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dot_Product_Formula\"><\/span>Dot Product Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The dot product is a mathematical formula that calculates the product of two vectors. The formula is:<\/p>\n<p>Vector a dot Vector b = |a| |b| cos(\u03b8)<\/p>\n<p>In this formula, a and b are the vectors, |a| is the magnitude of a, and |b| is the magnitude of b. \u03b8 is the angle between a and b.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dot_Product_Geometry_Definition\"><\/span>Dot Product Geometry Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In mathematics, the dot product is an operation that takes two vectors in a three-dimensional space and returns a single number. The dot product define as the sum of the products of the corresponding components of the vectors.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dot_Product_Algebra_Definition\"><\/span>Dot Product Algebra Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Dot Product Algebra Definition is a mathematical operation that calculates the product of two vectors. It represent by the symbol \u201c.\u201d and is usually calculate using the Cartesian coordinate system.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Properties_of_Dot_Product_of_Two_Vectors\"><\/span>Properties of Dot Product of Two Vectors<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In mathematics, the dot product is a binary operation that takes two vectors, u and v, and produces a scalar quantity, often denoted by \u3008u, v\u3009, that is the product of the magnitude of u and the magnitude of v, and the cosine of the angle between them.<\/p>\n<p>The dot product is distributive, meaning that \u3008u, (v + w)\u3009 = \u3008u, v\u3009 +<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Dot_Product_of_Vector-Valued_Functions\"><\/span>Dot Product of Vector-Valued Functions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let be a vector-valued function on an open subset of . The dot product of and is define as<\/p>\n<p>where is the norm of .<\/p>\n<p>We will show that is a linear function. That is, for any two vectors and and any real numbers and ,<\/p>\n<p>The proof is by induction on . The base case is easy, since<\/p>\n<p>The inductive step is also easy, since<\/p>\n<p>for all vectors and all real numbers .<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Examples\"><\/span>Solved Examples<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Question:<\/p>\n<p>What is the difference between an open and closed economy?<\/p>\n<p>An open economy is one in which the government allows foreigners to invest in the country and purchase its assets. A closed economy is one in which the government does not allow foreigners to invest in the country and purchase its assets.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Study_Material\"><\/span>NCERT Study Material<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT is a government organization which provides study material to students of classes I to XII. The NCERT study material is very helpful for students as it helps them in understanding the concepts taught in school in a better way.<\/p>\n<p>The NCERT study material is available in both Hindi and English. It is very well-organise and easy to understand. The NCERT study material includes text books, work books, teachers\u2019 manuals and activity books.<\/p>\n<p>The text books are written<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Scalar Dot Product of Two Vectors Scalar Dot Product Of Two Vectors: Dot Product of Two Vectors &#8211; In order [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Scalar Dot Product Of Two Vectors","_yoast_wpseo_title":"Scalar Dot Product Of Two Vectors | Definition, Formulas, Properties, FAQ","_yoast_wpseo_metadesc":"Scalar Dot Product Of Two Vectors: In order to understand the Dot product of two vectors, we need to first understand what a projection is?","custom_permalink":"study-materials\/maths\/scalar-dot-product-of-two-vectors\/"},"categories":[13,21],"tags":[8385,8043,8044,8042],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with 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