Explanation Given : The given vectors, A , B , and D . Calculation : Show the vector diagram of vector A × D , A × ( B + D ) , and A × D as in Figure 1. Show the vector diagram as in Figure 2. Express the position vector of A as follows: A = A x i + A y j + A z k (1) Here, the vector component along the x direction is i, the vector component along the y direction is j, and the vector component along the z direction is k. Express the position vector of B as follows: B = B x i + B y j + B z k (2) Express the position vector of D as follows: D = D x i + D y j + D z k (3) Determine the cross product of A × ( B + D ) as follows: A × ( B + D ) = | i j k A x A y A z B x + D x B y + D y B z + D z | (4) Determine the cross product of A × B as follows: A × B = | i j k A x A y A z B x B y B z | (5) Determine the cross product of A × B as follows: A × D = | i j k A x A y A z D x D y D z | (6) Conclusion : Simplify Equation (4) and it can be written as follows: A × ( B + D ) = | i j k A x A y A z B x + D x B y + D y B z + D z | = { i [ ( A y ) ( B z + D z ) − ( B y + D y ) ( A z ) ] − j [ ( A x ) ( B z + D z ) − ( B x + D x ) ( A z ) ] + k [ ( A x ) ( B y + D y ) − ( B x + D x ) ( A y ) ] } = { i [ ( A y B z + A y D z ) − ( B y A z + D y A z ) ] − j [ ( A x B z + A x D z ) − ( B x A z + D x A z ) ] + k [ ( B y A x + D y A x ) − ( B x A y + D x A y ) ] } Simplify the Equation, A × ( B + D ) = { ( A y B z − B y A z ) i − ( A x B z − B x A z ) j + ( B y A x − B x A y ) k+ ( A y D z − D y A z ) i − ( A x D z − D x A z ) j+ ( D y A x − D x A y ) k } Simplify Equation (5) and it can be written as follows: A × B = | i j k A x A y A z B x B y B z | = { i [ ( A y ) ( B z ) − ( B y ) ( A z ) ] − j [ ( A x ) ( B z ) − ( B x ) ( A z ) ] + k [ ( A x ) ( B y ) − ( B x ) ( A y ) ] } = { i [ ( A y B z ) − ( B y A z ) ] − j [ ( A x B z ) − ( B x A z ) ] + k [ ( B y A x ) − ( B x A y ) ]

Statics and Mechanics of Materials Plus Mastering Engineering with Pearson eText - Access Card Package (5th Edition)

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ISBN: 9780134301006

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 3.4, Problem 1P

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To prove the distributive law for the vector cross product A×(B+D)=(A×B)+(A×D).

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Chapter 3.4, Problem 12FP

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To prove the distributive law for the vector cross product A × ( B + D ) = ( A × B ) + ( A × D ) .

Solution Summary: The author explains how the distributive law for the vector cross product underset_Yes is proved.

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To prove the distributive law for the vector cross product A×(B+D)=(A×B)+(A×D).

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