{"id":13625,"date":"2020-12-10T10:39:01","date_gmt":"2020-12-10T05:09:01","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=13625"},"modified":"2020-12-10T12:22:53","modified_gmt":"2020-12-10T06:52:53","slug":"dot-product","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/dot-product\/","title":{"rendered":"Dot Product"},"content":{"rendered":"<h2>Dot Product<\/h2>\n<p><iframe loading=\"lazy\" title=\"Vectors - The Dot Product\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/98C7iv8OcnI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Scalar or Dot product<\/h3>\n<p><strong>(1) Scalar or Dot product of two vectors:<\/strong><br \/>\nIf a and b are two non-zero vectors and \u03b8\u00a0be the angle between them, then their scalar product (or dot product) is denoted by <strong>a.b<\/strong> and is defined as the scalar |a||b| cos\u00a0\u03b8\u00a0, where |a| and |b|\u00a0are modulii of a and b respectively and 0\u2264 \u03b8\u2264\u03c0. Dot product of two vectors is a scalar quantity.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129152 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-1.png\" alt=\"Dot Product 1\" width=\"245\" height=\"193\" \/><\/p>\n<p><strong>Angle between two vectors:<\/strong><br \/>\nIf a, b be two vectors inclined at an angle \u03b8, then a.b =\u00a0|a||b| cos\u00a0\u03b8.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129153 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-2.jpg\" alt=\"Dot Product 2\" width=\"477\" height=\"201\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-2.jpg 477w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-2-300x126.jpg 300w\" sizes=\"(max-width: 477px) 100vw, 477px\" \/><\/p>\n<p><strong>(2) Properties of scalar product:<\/strong><\/p>\n<ol>\n<li><strong>Commutativity:<\/strong> The scalar product of two vector is commutative i.e., a . b = b . a.<\/li>\n<li><strong>Distributivity of scalar product over vector addition:<\/strong> The scalar product of vectors is distributive over vector addition i.e.,<br \/>\n(a) a.(b + c) = a . b \u2212 a . c, (Left distrihutivity)<br \/>\n(b) (b + c).a = b . a + c . a , (Right distrihutivity)<\/li>\n<li>Let a and b be two non-zero vectors a . b = 0 \u21d4 a \u22a5 b.<br \/>\nAs i, j, k are mutually perpendicular unit vectors along the co-ordinate axes, therefore,\u00a0i . j = j . i =0 ; j . k = k . j = 0; k . i = i . k = 0.<\/li>\n<li>For any vector a, a . a = |a|<sup>2<\/sup>.<br \/>\nAs i, j, k are unit vectors along the co-ordinate axes, therefore i . i\u00a0= |i|<sup>2<\/sup>, j\u00a0. j\u00a0= |j|<sup>2<\/sup>\u00a0and\u00a0k\u00a0. k\u00a0= |k|<sup>2<\/sup><\/li>\n<li>If m, n are scalars and a, b be two vectors, then\u00a0ma . nb = mn(a . b) = (mn a).b = a.(mn b)<\/li>\n<li>For any vectors a and b, we have<br \/>\n(a) a. (\u2212b) = \u2212 (a.b) = (\u2212a).b<br \/>\n(b) (\u2212a).( \u2212b) = a.b<\/li>\n<li>For any two vectors a and b, we have\n<ol>\n<li>|a + b|<sup>2<\/sup> = |a|<sup>2<\/sup> + |b|<sup>2<\/sup> + 2a.b<\/li>\n<li>|a \u2212 b|<sup>2<\/sup> = |a|<sup>2<\/sup> + |b|<sup>2<\/sup> + 2a.b<\/li>\n<li>(a+b).(a\u2014b) = |a|<sup>2<\/sup> \u2212 |b|<sup>2<\/sup><\/li>\n<li>|a + b| =\u00a0 |a| + |b| \u21d2 a \u2225 b<\/li>\n<li>|a + b|<sup>2<\/sup> = |a|<sup>2<\/sup> + |b|<sup>2<\/sup> \u21d2 a \u22a5 b<\/li>\n<li>|a + b| = |a \u2212 b| = \u21d2 a \u22a5 b<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><strong>(3) Scalar product in terms of components:<\/strong><br \/>\nIf a = a<sub>1<\/sub>i + a<sub>2<\/sub>j + a<sub>3<\/sub>k and b = b<sub>1<\/sub>i + b<sub>2<\/sub>j + b<sub>3<\/sub>k then\u00a0a . b = a<sub>1<\/sub>b<sub>1<\/sub> + a<sub>2<\/sub>b<sub>2<\/sub> + a<sub>3<\/sub>b<sub>3<\/sub>.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129154\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-3.jpg\" alt=\"Dot Product 3\" width=\"600\" height=\"146\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-3.jpg 600w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-3-300x73.jpg 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><br \/>\nThe components of b along and perpendicular to are and respectively.<\/p>\n<p><strong>(4) Work done by a force:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129155\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-4.jpg\" alt=\"Dot Product 4\" width=\"535\" height=\"231\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-4.jpg 535w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-4-300x130.jpg 300w\" sizes=\"(max-width: 535px) 100vw, 535px\" \/><br \/>\nIf a number of forces are acting on a particle, then the sum of the works done by the separate forces is equal to the work done by the resultant force.<\/p>\n<h3>Scalar triple product<\/h3>\n<p><strong>(1) Scalar triple product of three vectors:<\/strong><br \/>\nIf a, b, c are three vectors, then their scalar triple product is defined as the dot product of two vectors a and b \u00d7 c. It is generally denoted by a . (b \u00d7 c) or [a b c].<\/p>\n<p><strong>(2) Properties of scalar triple product<\/strong>:<\/p>\n<ol>\n<li>If a, b, c are cyclically permuted, the value of scalar triple product remains the same. i.e., (a \u00d7 b). c = (b \u00d7 c). a = (c \u00d7 a). b or [a b c] = [b c a] = [c a b]<\/li>\n<li>The change of cyclic order of vectors in scalar triple product changes the sign of the scalat\u2019 triple product but not the magnitude i.e., [a b c] = \u2212[b a c] = \u2212[c b a] = \u2212[a c b]<\/li>\n<li>In scalar triple product the positions of dot and cross can be interchanged provided that the cyclic order of the vectors remains same i.e.,\u00a0(a \u00d7 b). c = a. (b \u00d7 c)<\/li>\n<li>The scalar triple product of three vectors is zero if any two of them are equal.<\/li>\n<li>For any three vectors a, b, c\u00a0and scalar \u03bb, [\u03bb a b c] =\u00a0\u03bb[a b c]<\/li>\n<li>The scalar triple product of three vectors is zero if any two of them are parallel or collinear.<\/li>\n<li>If a, b, c, d are four vectors, then\u00a0[(a + b) c d] = [a c d] + [b c d].<\/li>\n<li>The necessary and sufficient condition for three non-zero non-collinear vectors to be coplanar is that [a b c] = 0.<\/li>\n<li>Four points with position vectors a, b, c and d will be coplanar, if [ a b c] + [d c a] + [d a b] = [a b c].<\/li>\n<li>Volume of parallelopiped whose coterminous edges are a, b, c is [a b c] or a(b \u00d7 c).<\/li>\n<\/ol>\n<p><strong>(3) Scalar triple product in terms of components:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129157 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-5.jpg\" alt=\"Dot Product 5\" width=\"533\" height=\"502\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-5.jpg 533w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-5-300x283.jpg 300w\" sizes=\"(max-width: 533px) 100vw, 533px\" \/><br \/>\n<strong>(4) Tetrahedron:<\/strong><br \/>\nA tetrahedron is a three-dimensional figure formed by four triangle OABC is a tetrahedron with \u2206ABC as the base. OA, OB, OC, AB, BC and CA are known as edges of the tetrahedron. OA, BC; OB, CA and OC, AB are known as the pairs of opposite edges. A tetrahedron in which all edges are equal, is called a regular tetrahedron. Any two edges of regular tetrahedron are perpendicular to each other.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129158 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-6.jpg\" alt=\"Dot Product 6\" width=\"226\" height=\"204\" \/><\/p>\n<p><strong>Volume of tetrahedron<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129159\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-7.jpg\" alt=\"Dot Product 7\" width=\"440\" height=\"128\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-7.jpg 440w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-7-300x87.jpg 300w\" sizes=\"(max-width: 440px) 100vw, 440px\" \/><\/p>\n<ol>\n<li>The volume of a tetrahedron<\/li>\n<li>If a, b, c are position vectors of vertices A, B and C with respect to O, then volume of tetrahedron OABC = \\(\\frac { 1 }{ 6\u00a0}\\) [a b c].<\/li>\n<li>If a, b, c, d are position vectors of vertices A, B, C, D of a tetrahedron ABCD, then its volume = \\(\\frac { 1 }{ 6\u00a0}\\) [b\u2212a c\u2212a\u00a0d\u2212a].<\/li>\n<\/ol>\n<p><strong>(5) Reciprocal system of vectors:<\/strong><br \/>\nLet be three non-coplanar vectors, and let <img loading=\"lazy\" class=\"alignnone size-full wp-image-129160\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-8.jpg\" alt=\"Dot Product 8\" width=\"331\" height=\"62\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-8.jpg 331w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-8-300x56.jpg 300w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/>. a<em>&#8216;<\/em>, b<em>&#8216;<\/em>, c<em>&#8216;<\/em> are said to form a reciprocal system of vectors for the vectors a, b, c.<br \/>\nIf a, b, c and a<em>&#8216;<\/em>, b<em>&#8216;<\/em>, c<em>&#8216;<\/em>\u00a0form a reciprocal system of vectors, then<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129161\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-9.jpg\" alt=\"Dot Product 9\" width=\"514\" height=\"178\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-9.jpg 514w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-9-300x104.jpg 300w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><\/p>\n<h3>Scalar product of four vectors<\/h3>\n<p>(a \u00d7 b) . (c \u00d7 d) is a scalar product of four vectors. It is the dot product of the vectors a \u00d7 b and c \u00d7 d.<br \/>\nIt is a scalar triple product of the vectors a,b and c \u00d7 d as well as scalar triple product of the vectors a \u00d7 b, c and d.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129162\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Dot-Product-10.jpg\" alt=\"Dot Product 10\" width=\"292\" height=\"69\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dot Product Scalar or Dot product (1) Scalar or Dot product of two vectors: If a and b are two non-zero vectors and \u03b8\u00a0be the angle between them, then their scalar product (or dot product) is denoted by a.b and is defined as the scalar |a||b| cos\u00a0\u03b8\u00a0, where |a| and |b|\u00a0are modulii of a and &#8230; <a title=\"Dot Product\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/dot-product\/\" aria-label=\"More on Dot Product\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[5055,5072,5075,5079,5081,5073,5074,5076,5077,5080,5078],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Dot Product - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cbselibrary.com\/dot-product\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dot Product\" \/>\n<meta property=\"og:description\" content=\"Dot Product Scalar or Dot product (1) Scalar or Dot product of two vectors: If a and b are two non-zero vectors and \u03b8\u00a0be the angle between them, then their scalar product (or dot product) is denoted by a.b and is defined as the scalar |a||b| cos\u00a0\u03b8\u00a0, where |a| and |b|\u00a0are modulii of a and ... 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