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The Unit Vector

Unit Vector in 3D

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 green unit vector is pointing straight out of the screen, towards you. This green unit vector is perpendicular to both the blue and red unit vectors. What this means is the green unit vector is exactly 90 degrees away from the red and blue unit vectors. This makes the green unit vector posses a third dimension, meaning that unit vectors can exist in 3D. The black unit vector can rotate in any direction and angle. Below are some directions to follow to changing these attributes. When you start changing the angle of rotation for the black unit vector notice that whatever angle of rotation you set the unit vector to follow, it will always rotate like a circle, about the origin (0, 0). Here is another connection you can see with mathematics, noticing that a unit vector in 3D not only has magnitude (length) 1, but that this is the unit sphere.

Gray Vector:

x, y, z:

Purple Vector:

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

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

and

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

or

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

or

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

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〉,〈0, 1, 0〉and〈0, 0, 1〉each of magnitude 1. So the equation now becomes:

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

and

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

and

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

Here the û normalized vectors are all equal to 1. This is the case for all unit vectors.

You have now learned about the unit vector. Go back now to the vector page to move onto the next lesson.

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