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The Unit Vector

Unit Vector in 2D

A unit vector is a vector with a magnitude (length) of 1.

In the simulation below, you can see three unit vectors, all of magnitude (length) 1. The direction and angle of these unit vectors however is different. The blue unit vector is pointing to the north with an angle of 90 degrees, the red unit vector is pointing to the east with an angle of 0, and the black unit vector can be pointed in any direction and have any angle, based on the movement of your mouse. Notice that when you move your mouse very quickly around the origin (0, 0), the black unit vector goes around in a circle. Here you can see a connection with mathematics as this is not only a vector with a of magnitude (length) 1, but that this is the unit circle.

A unit vector can be represented in numerous ways. Here are the three most common ways of portraying them:

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 0 \end{bmatrix} \]

or

\[ \vec{A} = 1\hat{i} + 0\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle 1, 0, 0 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & 0 \end{bmatrix} \]

or

\[ \vec{B} = 0\hat{i} + 1\hat{j} + 0\hat{k} \]

or

\[ \vec{B} = \langle 0, 1, 0 \rangle \]

A unit vector can be calculated using the folliwng equation:

\[ \hat{u} = \frac{u}{|\text{u}|} \]

û is the normalized vector, u is a non-zero vector, and |u| is the magnitude (length) of the vector u.

In this example, the vectors are non-zero vectors〈1, 0, 0〉and〈0, 1, 0〉both of magnitude 1. So the equation now becomes:

\[ \hat{u} = \frac{\langle 1, 0, 0 \rangle}{|1|} = 1 \]

and

\[ \hat{u} = \frac{\langle 0, 1, 0 \rangle}{|1|} = 1 \]

In the examples above, we only considered vectors in the positive direction. Even if you move the black unit vector opposite the red unit vector so that it is pointing in the negative direction, the magnitude will still be a positive 1.

\[ \hat{u} = \frac{\langle -1, 0, 0 \rangle}{|1|} = 1 \]

As you can see the û normalized vectors are still equal to 1. This is the case for all unit vectors. It does not matter whether the unit vector is pointing in the negative or positive direction or even has a different angle, the unit vector will always be 1.

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