   Chapter 12.3, Problem 58E

Chapter
Section
Textbook Problem

# If c = | a | b + | b | a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.

To determine

To show: The c bisects the angle between a and b.

Explanation

Given:

c=|a|b+|b|a .

Here,

a,b and c are all nonzero vectors.

Formula:

Write the expression to find ab in terms of θ .

ab=|a||b|cosθ

Here,

|a| is the magnitude of a vector,

|b| is the magnitude of b vector, and

θ is the angle between vectors a and b.

Rearrange equation.

cosθ=ab|a||b| (1)

Consider α be the angle between a and c vectors and β be the angle between c and b vectors.

Rearrange equation (1) for angle α .

cosα=ac|a||c|

Substitute |a|b+|b|a for c .

cosα=a(|a|b+|b|a)|a||c|=a|a|b+a|b|a|a||c|=a|a|b+|a|2|b||a||c|=|a|(ab+|a||b|)|a||c|

cosα=ab+|a||b||c| (1)

Rearrange equation (1) for angle β

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