{"id":190320,"date":"2025-02-12T06:20:38","date_gmt":"2025-02-12T06:20:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=190320"},"modified":"2025-02-12T06:20:40","modified_gmt":"2025-02-12T06:20:40","slug":"levi-civita-symbol-and-bac-cab-rule","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/12\/levi-civita-symbol-and-bac-cab-rule\/","title":{"rendered":"Levi-Civita Symbol And BAC-CAB Rule"},"content":{"rendered":"\n

Levi-Civita Symbol And BAC-CAB Rule. Prove The BAC – CAB Rule A Times (B Times C) = B (A Middot C) – C (A Middot B) Using Summation Notation And The Levi-Civita Symbol. Epsilon_i Jk = {1 For I Jk = 123, 312, Or 231 -1 For I Jk = 213, 321, Or 132 0 Otherwise Note That An Identity Of The Levi-Civita Tensor Will Be Useful Sigma_k = 1^3 Epsilon_i Jk<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To prove the BAC-CAB rule (\\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) = \\mathbf{B} ( \\mathbf{A} \\cdot \\mathbf{C}) – \\mathbf{C} ( \\mathbf{A} \\cdot \\mathbf{B})) using summation notation and the Levi-Civita symbol, we will first express the cross products and dot products in terms of summation notation and the Levi-Civita symbol.<\/p>\n\n\n\n

Step 1: Express Cross Products and Dot Products Using Levi-Civita Symbol<\/h3>\n\n\n\n

We know the cross product can be written as:<\/p>\n\n\n\n

[
\\mathbf{A} \\times \\mathbf{B} = \\epsilon_{ijk} A_j B_k \\hat{e}i ] where (\\epsilon<\/em>{ijk}) is the Levi-Civita symbol, and (\\hat{e}_i) is the unit vector in the (i)-th direction. Similarly, we can write the dot product between two vectors as:<\/p>\n\n\n\n

[
\\mathbf{A} \\cdot \\mathbf{B} = A_i B_i
]<\/p>\n\n\n\n

Step 2: Express ( \\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) ) Using Levi-Civita Symbol<\/h3>\n\n\n\n

Now, let’s compute ( \\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) ) in summation notation.<\/p>\n\n\n\n

Using the identity for the cross product:<\/p>\n\n\n\n

[
\\mathbf{B} \\times \\mathbf{C} = \\epsilon_{ijk} B_j C_k \\hat{e}_i
]<\/p>\n\n\n\n

Now, take the cross product of (\\mathbf{A}) with this result:<\/p>\n\n\n\n

[
\\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) = \\epsilon_{ilm} A_l \\left( \\epsilon_{ijk} B_j C_k \\right) \\hat{e}_i
]<\/p>\n\n\n\n

Step 3: Simplify Using the Levi-Civita Identity<\/h3>\n\n\n\n

We can simplify the expression by using the identity for the contraction of two Levi-Civita symbols:<\/p>\n\n\n\n

[
\\epsilon_{ilm} \\epsilon_{ijk} = \\delta_{il} \\delta_{mk} – \\delta_{im} \\delta_{lk}
]<\/p>\n\n\n\n

This identity allows us to simplify the product of the two Levi-Civita symbols:<\/p>\n\n\n\n

[
\\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) = \\left( \\delta_{il} \\delta_{mk} – \\delta_{im} \\delta_{lk} \\right) A_l B_j C_k \\hat{e}_i
]<\/p>\n\n\n\n

Expanding the terms:<\/p>\n\n\n\n

[
\\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) = A_m (B_j C_k) \\hat{e}_m – A_l (C_k B_j) \\hat{e}_l
]<\/p>\n\n\n\n

Step 4: Rearranging Terms to Achieve the BAC-CAB Rule<\/h3>\n\n\n\n

Now, let\u2019s look at the expression:<\/p>\n\n\n\n

[
\\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) = \\mathbf{B} (\\mathbf{A} \\cdot \\mathbf{C}) – \\mathbf{C} (\\mathbf{A} \\cdot \\mathbf{B})
]<\/p>\n\n\n\n

By recognizing that (\\mathbf{A} \\cdot \\mathbf{B} = A_l B_l) and (\\mathbf{A} \\cdot \\mathbf{C} = A_k C_k), we arrive at the BAC-CAB identity:<\/p>\n\n\n\n

[
\\mathbf{A} \\times (\\mathbf{B} \\times \\mathbf{C}) = \\mathbf{B} (\\mathbf{A} \\cdot \\mathbf{C}) – \\mathbf{C} (\\mathbf{A} \\cdot \\mathbf{B})
]<\/p>\n\n\n\n

Thus, we have proven the BAC-CAB rule using the Levi-Civita symbol.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The BAC-CAB rule is a vector identity that simplifies the cross product of a vector with the cross product of two other vectors. The use of summation notation and the Levi-Civita symbol helps to break down the cross and dot products in terms of components, which can be manipulated algebraically. The Levi-Civita symbol provides a way to handle the orientation and antisymmetry of the cross product, and the contraction identity allows the combination of these terms into a simpler form. This approach avoids directly using geometric intuition and instead relies on the algebraic properties of the cross and dot products.<\/p>\n","protected":false},"excerpt":{"rendered":"

Levi-Civita Symbol And BAC-CAB Rule. Prove The BAC – CAB Rule A Times (B Times C) = B (A Middot C) – C (A Middot B) Using Summation Notation And The Levi-Civita Symbol. Epsilon_i Jk = {1 For I Jk = 123, 312, Or 231 -1 For I Jk = 213, 321, Or 132 0 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-190320","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=190320"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190320\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}