{"id":13599,"date":"2020-12-10T07:23:03","date_gmt":"2020-12-10T01:53:03","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=13599"},"modified":"2020-12-10T12:19:26","modified_gmt":"2020-12-10T06:49:26","slug":"cross-product","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/cross-product\/","title":{"rendered":"Cross Product"},"content":{"rendered":"<h2>Cross Product<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/h0NJK4mEIJU?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Vector or Cross product<\/h3>\n<p><strong>(1) Vector product of two vectors: <\/strong><br \/>\nLet a, b be two non-zero, non-parallel vectors.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-129150 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-1.png\" alt=\"Cross Product 1\" width=\"480\" height=\"375\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-1.png 480w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-1-300x234.png 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/>Then a \u00d7 b = |a||b| sin \u03b8 \\(\\hat { n }\\), and a \u00d7 b = |a||b| sin \u03b8 where \u03b8 is the angle between a and b, \\(\\hat { n }\\)\u00a0is a unit vector perpendicular to the plane of a and b such that a, b, \\(\\hat { n }\\)\u00a0form a right-handed system.<\/p>\n<p><strong>(2) Properties of vector product<\/strong><\/p>\n<ol>\n<li>Vector product is not commutative i.e., if and b are any two vectors, then a \u00d7 b \u2260 b \u00d7 a , however, a \u00d7 b = \u2212(b \u00d7 a)<\/li>\n<li>If a, b are two vectors and m, n are scalars, then ma \u00d7 nb = mn(a \u00d7 b) = m(a \u00d7 nb) = n(ma \u00d7 b).<\/li>\n<li>Distributivity of vector product over vector addition.<br \/>\nLet a, b, c be any three vectors. Then<br \/>\n(a) a \u00d7 (b + c) = a \u00d7 b + a \u00d7 c (Left distrihutivity).<br \/>\n(b) (b + c) \u00d7 a = b \u00d7 a + c \u00d7 a (Right distributivity).<\/li>\n<li>For any three vectors a, b, c\u00a0we have a\u00d7(b \u2212 c) = a\u00d7b \u2212 a\u00d7c.<\/li>\n<li>The vector product of two non-zero vectors is zero vector iff they are parallel (Collinear) i.e. a \u00d7 b = 0 \u21d4 a \u2225 b, a, b are non-zero vectors.<br \/>\nIt follows from the above property that a \u00d7 a = 0\u00a0for every non-zero vector a, which in turn implies that i \u00d7 i = j \u00d7 j = k \u00d7 k = 0.<\/li>\n<li>Vector product of orthonormal triad of unit vectors i, j, k using the definition of the vector product, we obtain i \u00d7 j = k, j \u00d7 k = i, k \u00d7 i = j, j \u00d7 i = \u2212k, k \u00d7 j = \u2212i, i \u00d7 k = \u2212j.<\/li>\n<\/ol>\n<p><strong>(3) Vector 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.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-129144 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-2.jpg\" alt=\"Cross Product 2\" width=\"572\" height=\"334\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-2.jpg 572w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-2-300x175.jpg 300w\" sizes=\"(max-width: 572px) 100vw, 572px\" \/><br \/>\n<strong>(4) Angle between two vectors:<\/strong><br \/>\nIf \u03b8\u00a0is the angle between a and b,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-129145 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-3.jpg\" alt=\"Cross Product 3\" width=\"208\" height=\"68\" \/><br \/>\n<strong>(5) (i) Right handed system of vectors:<\/strong><br \/>\nThree mutually perpendicular vectors a, b, c form a right handed system of vector iff a \u00d7 b = c, b \u00d7 c = a, c \u00d7 a = b.<br \/>\nExamples: The unit vectors i, j, k form a right-handed system,\u00a0i \u00d7 j = k, j \u00d7 k = i, k \u00d7 i = j.<br \/>\n<strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-129147 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-4.jpg\" alt=\"Cross Product 4\" width=\"535\" height=\"252\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-4.jpg 535w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-4-300x141.jpg 300w\" sizes=\"(max-width: 535px) 100vw, 535px\" \/>(ii) Left handed system of vectors:<\/strong> The vectors a, b, c\u00a0mutually perpendicular to one another form a left handed system of vector iff\u00a0c \u00d7 b = a, a \u00d7 c = b, b \u00d7 a = c.<br \/>\n<strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-129148 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-5.jpg\" alt=\"Cross Product 5\" width=\"254\" height=\"195\" \/>(6) Vector normal to the plane of two given vectors:<\/strong><br \/>\nIf a, b be two non-zero, nonparallel vectors and let \u03b8\u00a0be the angle between them. a \u00d7 b = |a||b| sin \u03b8 \\(\\hat { n }\\) where \\(\\hat { n }\\) is a unit vector perpendicular to the plane of a and b such that a, b, \\(\\hat { n }\\)\u00a0form a right-handed system.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-129149\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-6.jpg\" alt=\"Cross Product 6\" width=\"657\" height=\"305\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-6.jpg 657w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Cross-Product-6-300x139.jpg 300w\" sizes=\"(max-width: 657px) 100vw, 657px\" \/><br \/>\n<strong>(7) Area of parallelogram and triangle:<\/strong><\/p>\n<ol>\n<li>The area of a parallelogram with adjacent sides a and b is |a \u00d7 b|.<\/li>\n<li>The area of a parallelogram with diagonals d<sub>1<\/sub> and d<sub>2<\/sub> is \\(\\frac { 1 }{ 2 }\\) |d<sub>1<\/sub>\u00d7 d<sub>2<\/sub>|.<\/li>\n<li>The area of a plane quadrilateral ABCD is \\(\\frac { 1 }{ 2 } \\left| \\overrightarrow { AC } \\times \\overrightarrow { BD } \u00a0\\right|\\), where AC and BD are its diagonals.<\/li>\n<li>The area of a triangle with adjacent sides a and b is \\(\\frac { 1 }{ 2 }\\)\u00a0|a \u00d7 b|<\/li>\n<li>The area of a triangle ABC is \\(\\frac { 1 }{ 2 } \\left| \\overrightarrow { AB } \\times \\overrightarrow { AC\u00a0} \u00a0\\right|\\)\u00a0or \\(\\frac { 1 }{ 2 } \\left| \\overrightarrow { BC\u00a0} \\times \\overrightarrow { BA } \u00a0\\right|\\)\u00a0or\u00a0\\(\\frac { 1 }{ 2 } \\left| \\overrightarrow { CB\u00a0} \\times \\overrightarrow { CA\u00a0} \u00a0\\right|\\).<\/li>\n<li>If a, b, c are position vectors of \u2206ABC vertices of a then its area = \\(\\frac { 1 }{ 2 }\\)\u00a0|(a \u00d7 b) + (b \u00d7 c) + (c \u00d7 a)|<\/li>\n<li>Three points with position vectors a, b, c\u00a0are collinear if (a \u00d7 b) + (b \u00d7 c) + (c \u00d7 a) = 0..<\/li>\n<\/ol>\n<p><strong>(8) Moment of a force:<\/strong><br \/>\nMoment of a force F about a point O is \\(\\overrightarrow { OP } \\times F\\), where P is any point on the line of action of the force F.<\/p>\n<h3>Vector triple product<\/h3>\n<p>Let a, b, c he any three vectors, then the vectors a \u00d7 (b \u00d7 c) and (a \u00d7 b) \u00d7 c are called vector triple product of a, b, c .<br \/>\nThus, a \u00d7(b \u00d7 c) = (a. c) b \u2014 (a . b) c<\/p>\n<p><strong>Properties of vector triple product:<\/strong><\/p>\n<ol>\n<li>The vector triple product a \u00d7 (b \u00d7 c) is a linear combination of those two vectors which are within brackets.<\/li>\n<li>The vector r = a \u00d7(b \u00d7 c) is perpendicular to a and lies in the plane of b and c.<\/li>\n<li>The formula a \u00d7(b \u00d7 c) = (a.c)b \u2212 (a.b)c is true only when the vector outside the bracket is on the left most side. If it is not, we first shift on left by using the properties of cross product and<br \/>\nthen apply the same formula.<br \/>\nThus, (b \u00d7 c)\u00d7 a = \u2212 {a \u00d7(b \u00d7 c)} = \u2212 {(a.c)b \u2212 (a.b)c} = (a.b)c \u2212 (a.c)b<\/li>\n<li>Vector triple product is a vector quantity.<\/li>\n<li>a \u00d7 (b \u00d7 c) \u2260 (a \u00d7 b)\u00d7 c.<\/li>\n<\/ol>\n<h3>Vector product of four vectors<\/h3>\n<ol>\n<li>(a \u00d7 b ) \u00d7 (c \u00d7 d) is a vector product of four vectors.<br \/>\nIt is the cross product of the vectors a \u00d7 b and c \u00d7 d.<\/li>\n<li>a\u00d7{b \u00d7(c \u00d7d)}, {(a\u00d7b)\u00d7c}\u00d7d are also different products of four vectors a, b, c and d.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Cross Product Vector or Cross product (1) Vector product of two vectors: Let a, b be two non-zero, non-parallel vectors. Then a \u00d7 b = |a||b| sin \u03b8 , and a \u00d7 b = |a||b| sin \u03b8 where \u03b8 is the angle between a and b, \u00a0is a unit vector perpendicular to the plane of [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[5055,5057,5053,5059,5056,5058,5051,5054,5061,5052,5060],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Cross Product - A Plus Topper<\/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:\/\/www.aplustopper.com\/cross-product\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cross Product\" \/>\n<meta property=\"og:description\" content=\"Cross Product Vector or Cross product (1) Vector product of two vectors: Let a, b be two non-zero, non-parallel vectors. 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