Unit Vector

A vector has magnitude (its length) and direction:

Vector arrow labeled with magnitude as length and direction as angle

Unit Vector

A Unit Vector has a magnitude of 1:

Vector arrow with a length of exactly 1 unit

Its symbol is usually a lowercase letter with a "hat", for example:

(Pronounced "a-hat")

Scaling

We can scale a unit vector by multiplying it by a number. Here, vector a is 2.5 times the unit vector. Notice it points in the same direction:

Comparison of a unit vector and a parallel vector that's 2.5 times longer

Finding a Unit Vector

To find a unit vector with the same direction as a given vector, we divide that vector by its magnitude (length):

v = v|v|

where:

v = unit vector along v
v = the vector
|v| = magnitude of v

Example: find the unit vector along v = (3, 4)

Magnitude |v| = √(3² + 4²) = √(9 + 16) = √25 = 5

Now divide each component by 5:

v = (35, 45) = (0.6, 0.8)

Check: its magnitude is √(0.6² + 0.8²) = √(0.36 + 0.64) = √1 = 1. It works!

In 2 Dimensions

Unit vectors can be used in 2 dimensions:

Vector on a grid shown as the sum of 2 horizontal and 1.3 vertical unit vectors

Here we show that the vector a is made up of 2 "x" unit vectors and 1.3 "y" unit vectors.

In this case we often use i and j like this:

i (for the x-axis)
j (for the y-axis)

So the vector a = 2i + 1.3j

Any Direction

A unit vector can point in any direction! While we often use them along an axis, a vector pointing at 45° with a length of 1 is still a unit vector.

In 3 Dimensions

We can also use unit vectors in three (or more!) dimensions:

Three perpendicular unit vectors on x, y, and z axes in a 3D coordinate system

For 3D we can use i (x-axis), j (y-axis) and k (z-axis).

Advanced topic: arranged like this the three unit vectors form a basis of 3D space. But that's not the only way to do this ... learn more at Matrix Rank.

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