   Chapter 12.3, Problem 18E

Chapter
Section
Textbook Problem

# Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)18. a = ⟨−1, 3, 4⟩, b = ⟨5, 2, 1⟩

To determine

To find: The angle between vectors a and b .

Explanation

Given:

a=1,3,4 , b=5,2,1 .

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|) (1)

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

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

ab=a1,a2,a2b1,b2,b2

ab=a1b1+a2b2+a3b3 (2)

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

|a|=a12+a22+a32 (3)

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

|b|=b12+b22+b32 (4)

In equation (2), substitute –1 for a1 , 3 for a2 , 4 for a3 , 5 for b1 , 2 for b2 and 1 for b3

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