   Chapter 12.3, Problem 19E

Chapter
Section
Textbook Problem

# Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)19. a = 4i −3j + k, b = 2i − k

To determine

To find: The angle between vectors a and b .

Explanation

Given:

a=4i3j+k (1)

b=2ik (2)

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 the equation.

cosθ=ab|a||b|

θ=cos1(ab|a||b|) (3)

A minimum of two vectors are required to form a dot product. The resultant dot product of two vectors is scalar. So, the dot product is also known as a scalar product.

Consider vectors a and b as follows.

a=a1i+a2j+a3k (4)

b=b1i+b2j+b3k (5)

Consider a general expression to find the dot product between two three-dimensional vectors.

ab=a1,a2,a2b1,b2,b2

ab=a1b1+a2b2+a3b3 (6)

Consider a general expression to find the magnitude of a three-dimensional vector that is a=a1,a2,a3 .

|a|=a12+a22+a32 (7)

Similarly, consider a general expression to find the magnitude of a three-dimensional vector that is b=b1,b2,b3 .

|b|=b12+b22+b32 (8)

Compare equation (1), equation (4) and equation (2), equation (5)

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