   Chapter 12.4, Problem 25E

Chapter
Section
Textbook Problem

# Prove the property of cross products (Theorem 11).25. Property 3: a × (b + c) = a × b + a × c

To determine

To prove: The Property 3: a×(b+c)=a×b+a×c .

Explanation

Formula used:

Consider the expression to find dot product between two vectors a=a1,a2,a3 and b=b1,b2,b3 .

(a×b)=|ijka1a2a3b1b2b3| (1)

Consider three three-dimensional vectors such as a=a1,a2,a3 , b=b1,b2,b3 and c=c1,c2,c3

Find a×b using equation (1).

(a×b)=|ijka1a2a3b1b2b3|=i|a2a3b2b3|j|a1a3b1b3|+k|a1a2a1b2|=i(a2b3b2a3)j(a1b3b1a3)+k(a1b2b1a2)=i(a2b3b2a3)+j(b1a3a1b3)+k(a1b2b1a2)

Modify the equation.

(a×b)=(a2b3b2a3),(b1a3a1b3),(a1b2b1a2)

Similarly find a×c using equation (1).

(a×c)=|ijka1a2a3c1c2c3|=i|a2a3c2c3|j|a1a3c1c3|+k|a1a2c1c2|=i(a2c3c2a3)j(a1c3c1a3)+k(a1c2c1a2)=i(a2c3c2a3)+j(c1a3a1c3)+k(a1c2c1a2)

Modify the equation.

(a×c)=(a2c3c2a3),(c1a3a1c3),(a1c2c1a2)

Find the cross product between a and (b+c)

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