Unit Vectors

Introduction: In this lesson, unit vectors and their basic components will be defined and quantified. We will examine both 2- and 3-dimensional vectors.

A unit vector is a vector which has a magnitude of 1.

For example, the vector v = (1, 3) is not a unit vector because

.

The notation represents the norm, or magnitude, of vector v.

But the vector w =

is a unit vector because

.

The basic unit vectors are i = (1, 0) and j = (0, 1) which are of length 1 and have directions along the positive x-axis and y-axis respectively.

To find a unit vector with the same direction as a given vector, we divide by the magnitude of the vector. For example, consider the vector v = (1, 3) which has a magnitude of . If we divide each component of v by

we will get the unit vector u_v which is in the same direction as v:

.

1. What is a unit vector which has the same direction as r = (2, – 5)?

We first calculate . We divide each component of r by and we have:

2. Describe the vector v = (3,– 2) using the basic unit vectors i and j.

We use scalar multiplication because 3i = (3, 0) and – 2j = (0, – 2) and we have v = 3i – 2j.

In three dimensions, we have i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1) where each of these vectors is a unit vector pointing in the direction of the positive x-, y-, and z-axes respectively.

3. Write vector v = (1, -3, 4) as a combination of the basic vectors i, j and k. What is a unit vector which has the same direction as v?

As in example (ii), we have v = i – 3j +4k.

To find a unit vector with the same direction as v we first calculate .

Therefore we have