7 Unit Vectors
We know vectors have a magnitude and a direction. The magnitude of a vector is just its length, so we can easily report that. But how can we specify the direction of a vector? To do so, we use a unit vector. Let 
be any vector. The unit vector 
is defined as
.
The “three line symbol” 
means is defined as.
Exercise 7.1: Unit Vectors
Compute the magnitude of the vector . Assuming the vector has units of meters, what are the units of ?
The fact that unit vectors have no units makes sense: unit vectors specify a direction, and directions have no units.
Given a coordinate system, there are a series of special unit vectors denoted by 
, 
, and 
, which point in the direction of the positive x-axis (), y-axis (), and z-axis (). The notation 
, 
, and 
is also common. Because both of these notations are standard, we will purposely use both notations throughout this book.
Exercise 7.2: Expressing Vectors in Terms of Unit Vectors
Use unit vectors to write the vector 
as a sum of two vectors, one proportional to , and one proportional to .
Evidently, writing 
is equivalent to writing 
. However, you will find as we progress in this book that the unit vector notation is much more useful. For this reason, from now on I will always use the unit vector notation.
To get used to this, and to practice a little bit of vector algebra, let’s do a few problems.
Exercise 7.3: Vector Algebra
Consider the vectors 
, 
, and 
.
Exercise 7.4: Using Unit Vectors
You are given the vectors 
and 
. Calculate the angle 
that the vector 
makes with the 
-axis. Report your result in radians, using two decimal places. You probably want to sketch this out!
Key Takeaways