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Chapter 9: Q. 37 (page 613)

Suppose that vand ware unit vectors. If the angle between vand iis 伪 and that between wand iis 尾, use the idea of the dot product v路wto prove that

cos伪-尾=cos伪cos尾+sin伪sin尾

Short Answer

Expert verified

It is proved thatcos伪-尾=cos伪cos尾+sin伪sin尾.

Step by step solution

01

Step 1. Given Information

there are two unit vectors vand w.The angle between vand iis 伪 and that between wand iis 尾. We have to use the idea of dot product and prove that
cos伪-尾=cos伪cos尾+sin伪sin尾.

02

Step 2. Proving

If vand ware unit vectors.

So, v=1,w=1.

Now, the dot product of

v路w=vw肠辞蝉胃v路w=肠辞蝉胃v=1,w=1

When 胃=伪-尾or尾-伪we get,

v路w=肠辞蝉胃v路w=cos伪-尾........(a)

Now,

v=v肠辞蝉伪i+蝉颈苍伪jv=肠辞蝉伪i+蝉颈苍伪jv=1

And

w=wcos尾i+sin尾jw=cos尾i+sin尾jw=1

So, cos伪-尾=v路wcos伪-尾=肠辞蝉伪i+蝉颈苍伪j路肠辞蝉尾i+蝉颈苍尾jcos伪-尾=肠辞蝉伪肠辞蝉尾+蝉颈苍伪蝉颈苍尾

Hence proved.

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