Angle Between Two Vectors and Vector Scalar Product

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.

A_x = 2; B_x = 1
A_y = 3; B_y = -2
A_z = 4; B_z = 3

The scalar product of two vectors is given by:

A · B = A B cos θ = |A||B| cos θ

or by:

A · B = A_xB_x + A_yB_y + A_zB_z

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

cos θ = (A_xB_x + A_yB_y + A_zB_z) / AB

For this problem:

A_xB_x + A_yB_y + A_zB_z = (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°