Chapter 9: Q. 62 (page 628)

Prove Property $(9) .$

Short Answer

Expert verified

We had proved $‖ u \times v ‖ = ‖ u ‖ ‖ v ‖ \sin θ .$

Step by step solution

Step 1. The value of u×v

We know that,

$(u \times v)^{2} = ‖ u ‖^{2} ‖ v ‖^{2} - (u \cdot v)^{2}$

Substituting $u \cdot v = ‖ u ‖ ‖ v ‖ \cos θ$

$= ‖ u ‖^{2} ‖ v ‖^{2} - (‖ u ‖ ‖ v ‖ \cos θ)^{2}$

$= ‖ u ‖^{2} ‖ v ‖^{2} - ‖ u ‖^{2} ‖ v ‖^{2} \cos^{2} θ$

Since, $(\sin^{2} θ + \cos^{2} θ = 1)$ , we can write the above as,

$= ‖ u ‖^{2} ‖ v ‖^{2} (1 - \cos^{2} θ)$

$= ‖ u ‖^{2} ‖ v ‖^{2} \sin^{2} θ$

Step 2. Deduce the equation

The equation obtained is :

$‖ u \times v ‖^{2} = ‖ u ‖^{2} ‖ v ‖^{2} \sin^{2} θ ‖ u \times v ‖ = ‖ u ‖ ‖ v ‖ \sin θ$

Hence Proved.

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Prove Property $(9) .$

Short Answer

Step by step solution

Step 1. The value of u×v

Step 2. Deduce the equation

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Prove Property(9).

Short Answer

Step by step solution

Step 1. The value of u×v

Step 2. Deduce the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Prove Property $(9) .$