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Chapter 9: Q. 62 (page 628)

Prove Property(9).

Short Answer

Expert verified

We had proved‖u×v‖=‖u‖‖v‖sinθ.

Step by step solution

01

Step 1. The value of u×v

We know that,

(u×v)2=‖u‖2‖v‖2-(u·v)2

Substitutingu·v=‖u‖‖v‖cosθ

=‖u‖2‖v‖2-(‖u‖‖v‖cosθ)2

=‖u‖2‖v‖2-‖u‖2‖v‖2cos2θ

Since, sin2θ+cos2θ=1, we can write the above as,

=‖u‖2‖v‖21-cos2θ

=‖u‖2‖v‖2sin2θ

02

Step 2. Deduce the equation

The equation obtained is :

‖u×v‖2=‖u‖2‖v‖2sin2θ‖u×v‖=‖u‖‖v‖sinθ

Hence Proved.

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