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Let u and v be two nonzero vectors in R2. Prove that uโ‹…v=โ€–uโ€–โ€–vโ€–cosฮธ where ฮธ is the angle between u and v.

Short Answer

Expert verified

It is proven that uโ‹…v=โ€–uโ€–โ€–vโ€–cosฮธ.

Step by step solution

01

Step 1. Prove

To prove that uโ‹…v=โ€–uโ€–โ€–vโ€–cosฮธ where ฮธ is the angle between u and v we will use the law of cosines and the properties of dot-product.

So, the law of cosines for the vectors u and v is

โ€–uโˆ’vโ€–2=โ€–uโ€–2+โ€–vโ€–2โˆ’2โ€–uโ€–โ€–vโ€–cosฮธ .....(a)

02

Step 2. Prove

Now, by using the dot-product we get,

โ€–uโˆ’vโ€–2=(uโˆ’v)โ‹…(uโˆ’v)

โ€–uโˆ’vโ€–2=uโ‹…uโˆ’uโ‹…v+vโ‹…u+vโ‹…v

โ€–uโˆ’vโ€–2=uโ‹…uโˆ’2uโ‹…v+vโ‹…v

โ€–uโˆ’vโ€–2=โ€–uโ€–2โˆ’2uโ‹…v+โ€–vโ€–2 .......(b)

By using equations (a) and (b)

โ€–uโ€–2โˆ’2uโ‹…v+โ€–vโ€–2=โ€–uโ€–2+โ€–vโ€–2โˆ’2โ€–uโ€–โ€–vโ€–cosฮธ

โˆ’2uโ‹…v=โˆ’2โ€–uโ€–โ€–vโ€–cosฮธ

uโ‹…v=โ€–uโ€–โ€–vโ€–cosฮธ

Hence proved.

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