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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.

The Lesson:
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 uv which is in the same direction as v:
.
Let's Practice:
  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:
  1. 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.
  1. 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

Examples
Example If vector v = (3, – 7) what is a unit vector with the same direction as v?
What is your answer?
 
Example Vector w = (2, 4, – 1). Rewrite w as a combination of the basic vectors i, j, and k. What is a unit vector with the same direction as w?
What is your answer?
 



M Ransom

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