   Chapter 11, Problem 15PS

Chapter
Section
Textbook Problem

# ProofConsider the vectors u = 〈 cos α , sin α , 0 〉 and v = 〈 cos β , sin β , 0 〉 where α > β . Find the cross product of the vectors and use the result to prove the identity sin ( α − β ) = sin α cos β − cos α sin β .

To determine

To prove: The identity sin(αβ)=sinαcosβcosαsinβ by using the cross product of the vectors u=cosα,sinα,0 and v=cosβ,sinβ,0 where α<β.

Explanation

Given:

The vectors given are u=cosα,sinα,0 and v=cosβ,sinβ,0 where α<β.

Formula used:

If u=u1i+u2j+u3k and v=v1i+v2j+v3k are vector in space. The cross product of u and v is

u×v=(u2v3u3v2)i-(u1v3u3v1)j+(u1v2u2v1)k

Proof:

Known that,

sin(αβ)=v×uuv

It is given that u=cosαi+sinαj+0k and v=cosβi+sinβj+0k.

∴, the values of u and v:

u=cos2α+sin2α+0=1

And,

v=cos2β+sin2β+0=1

Now, the cross product of u and v:

u×v=(u2v3u3v2)i-(u1v3u3v1)j+(u1v2

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