{"id":154526,"date":"2022-03-25T23:38:00","date_gmt":"2022-03-25T18:08:00","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/dot-product-of-two-vectors-properties-and-examples\/"},"modified":"2025-05-16T13:12:45","modified_gmt":"2025-05-16T07:42:45","slug":"dot-product-of-two-vectors-properties-and-examples","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/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\/maths\/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-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/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-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/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-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/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-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/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-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/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><\/ul><\/nav><\/div>\n<p><span style=\"font-size: 14pt;\"><strong>Dot Product of Two Vectors Class 11 &#8211; <\/strong><strong>Properties and Examples<\/strong><\/span><\/p>\n<p><strong><span style=\"color: #0000ff;\">Definition:<\/span> <\/strong>The<strong> dot product<\/strong> of two vectors is a scalar quantity that is computed by multiplying the magnitude of one vector by the magnitude of the other vector and then adding the products of the corresponding components.<\/p>\n<p>The dot product is commutative, meaning that the order of the vectors does not affect the result.<\/p>\n<p>The dot product is also associative, meaning that the order of the addition of the scalars does not affect the result.<\/p>\n<p>The dot product is distributive, meaning that the dot product of a vector and a scalar is the same as the scalar product of the vector and the scalar.<\/p>\n<p>The dot product is symmetric, meaning that the dot product of two<strong> vectors <\/strong>is the same as the dot product of the reversed vectors.<\/p>\n<p>The dot product is a real number.<\/p>\n<p>The dot product is positive if the vectors are pointing in the same direction and negative if the vectors are pointing in opposite directions.<\/p>\n<p>The dot product is zero if the vectors are perpendicular.<\/p>\n<p>The dot product is used in physics to calculate the magnitude and direction of a resultant force.<\/p>\n<p>The following example shows how to calculate the dot product of two vectors.<\/p>\n<p>Example:<\/p>\n<p>The vectors A = (4, -3) and B = (-5, 2) are given.<\/p>\n<p>The magnitude of A is 4 and the magnitude of B is 5.<\/p>\n<p>The dot product of A and B is 20.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154525 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/dot-product-of-two-vectors-properties-and-examples-300x212.jpg\" alt=\"Dot Product of Two Vectors\" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/dot-product-of-two-vectors-properties-and-examples-300x212.jpg?v=1648231674 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/dot-product-of-two-vectors-properties-and-examples.jpg?v=1648231674 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/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>The dot product between two vectors is a scalar quantity that is calculated by multiplying the corresponding components of the vectors and then summing them. The dot product is a measure of the similarity between the vectors and is used to determine the orientation of one vector relative to another.<\/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 of two vectors is a scalar quantity that is calculated by multiplying the magnitudes of the vectors and then multiplying the products of the corresponding components. The dot product is symbolized by the symbol &#8220;\u2022&#8221; and is usually denoted by the letter A.<\/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>The dot product is a measure of the similarity of two vectors. It is the product of the lengths of the vectors, multiplied by the cosine of the angle between them.<\/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 is a mathematical operation that takes two vectors and multiplies them together to produce a scalar. The result is a number that represents the magnitude of the vector sum of the two vectors.<\/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>The dot product of two vectors is a scalar quantity.<\/p>\n<p>The dot product is commutative.<\/p>\n<p>The dot product is distributive.<\/p>\n<p>The dot product is associative.<\/p>\n<p>The dot product is symmetric.<\/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>A dot product of vector-valued functions is a function of one variable, defined by<\/p>\n<p>(f \u2217 g)(x) = f(x)g(x)<\/p>\n<p>for all x in the domain of both f and g.<\/p>\n<p><span style=\"color: #ff0000;\"><strong>Also Read: <\/strong><\/span><\/p>\n<p><strong><a href=\"https:\/\/infinitylearn.com\/surge\/maths\/types-of-vectors\/\" target=\"_blank\" rel=\"noopener\">Types of Vectors <\/a><\/strong><\/p>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/maths\/coplanar-vectors\/\" target=\"_blank\" rel=\"noopener\"><strong>Coplanar Vectors <\/strong><\/a><\/p>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/maths\/addition-of-vectors\/\" target=\"_blank\" rel=\"noopener\"><strong>Addition of Vectors <\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dot Product of Two Vectors Class 11 &#8211; Properties and Examples Definition: The dot product of two vectors is a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Dot Product of Two Vectors class 11","_yoast_wpseo_title":"Dot Product of Two Vectors class 11 | Properties and Examples","_yoast_wpseo_metadesc":"In order to understand 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