   Chapter 11.4, Problem 58E

Chapter
Section
Textbook Problem

# Proof Prove that u × (   v × w ) = ( u ⋅ w ) v − ( u ⋅ v ) w .

To determine

To prove: The identity u×(v×w)=(uw)v(uv)w.

Explanation

Given:

The identity u×(v×w)=(uw)v(uv)w.

Proof:

Consider the identity,

u×(v×w)=(uw)v(uv)w

Let vectors u,v and w are given as:

u=a1,b1,c1=a1i+b1j+c1kv=a2,b2,c2=a2i+b2j+c2kw=a3,b3,c3=a3i+b3j+c3k

First, calculate the left- side of identity,

v×w=|ijka2b2c2a3b3c3|=i(b2c3b3c2)j(a2c3a3c2)+k(a2b3a3b2)

Now take the cross product of vector v×w with the vector u that is,

u×(v×w)=|ijka1b1c1b2c3b3c2a2c3a3c2a2b3a3b2|=[b1(a2b3a3b2)c1(a2c3a3c2)]i[a1(a2b3a3b2)c1(b2c3b3c2)]j+[a1(a2c3a3c2)b1(b2c3b3c2)]k=[a2(b1b3c1c3)a3(b1b2c1c2)]i+[b2(a1a3+c1c3)b3(a1a2+c1c2)]j+[c2(a1a3b1b3)+c3(a1a2b1b2)]k

Now consider the right-side of the identity,

(uw)v=(a1a3+b1b3+c1c3)(a2i+b2j+c2k)=[a

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