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# Angle Between Two Vectors and Vector Scalar Product

## Worked Vector Example Problem

This is a graphical representation of the angle between vectors.

Acdx, public domain
This is a worked example problem that shows how to find the angle between two vectors.

### Vector Problem

Find the angle between the two vectors:

A = 2i + 3j + 4k
B = i - 2j + 3k

### Solution

Write the components of each vector.

Ax = 2; Bx = 1
Ay = 3; By = -2
Az = 4; Bz = 3

The scalar product of two vectors is given by:

A · B = A B cos θ = |A||B| cos θ

or by:

A · B = AxBx + AyBy + AzBz

When you set the two equations equal and rearrange the terms you find:

cos θ = (AxBx + AyBy + AzBz) / AB

For this problem:

AxBx + AyBy + AzBz = (2)(1) + (3)(-2) + (4)(3) = 8

A = (22 + 32 + 42)1/2 = (29)1/2

B = (12 + (-2)2 + 32)1/2 = (14)1/2

cos θ = 8 / [(29)1/2 * (14)1/2] = 0.397

θ = 66.6°

Anne Marie Helmenstine, Ph.D.