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Chapter 10: Q. 5 (page 823)

If u and v are nonzero vectors in 鈩�3, what is the geometric relationship between u,v and u脳v?

Short Answer

Expert verified

The relation between u,vandu脳vis u脳v=u vsin胃,where 胃 is the angle between u and v.

Step by step solution

01

Step 1. Given Information 

If u and v are nonzero vectors in 鈩�3, what is the geometric relationship between u,v and u脳v?

02

Step 2. u and v are nonzero vectors in ℝ3 with the same initial point.

By Lagrange鈥檚 identity

u脳v2=u2v2鈭�(u路v)2

But since u路v=uvcos胃, we have

u脳v2=u2v2鈭�(uvcos胃)2u脳v2=u2v2(1-cos2胃)u脳v2=u2v2sin2胃

03

Step 3. Then the relation between u,v and u×v is 

u脳v=uvsin胃

where 胃 is the angle between u and v.

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