Unit Vector from a Vector

Preserving Direction with a Vector of Length One

A unit vector has length or magnitude $1 \text{ unit}$ . It represents direction without carrying the original magnitude. From any nonzero vector, we obtain a unit vector in the same direction by dividing the vector by its length.

Unit Vector Formula and Direction

Consider vector $\overrightarrow{PQ}$ in a Cartesian coordinate system. The unit vector of $\overrightarrow{PQ}$ is defined as vector $\overrightarrow{PQ}$ divided by its length.

\mathbf{u}_{PQ} = \frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}

Where $|\overrightarrow{PQ}|$ is the length or magnitude of vector $\overrightarrow{PQ}$ .

A unit vector always points in the same direction as the original vector but has its length normalized to $1 \text{ unit}$ .

Vector and Unit Vector

Visualization of the original vector and the unit vector that have the same direction but different lengths.

Calculating a Unit Vector

Dividing a vector by its own length keeps its direction and scales its length down to one. Work through the three steps below for any nonzero vector.

Identify the original vector
Calculate the length of the vector
Divide the vector by its length

Normalizing a Three Dimensional Vector

Take the vector $\vec{v} = (3, 6, 4)$ .

First, calculate its magnitude:

\begin{aligned} |\vec{v}| &= \sqrt{3^2 + 6^2 + 4^2} \\ &= \sqrt{9 + 36 + 16} \\ &= \sqrt{61} \end{aligned}

Then divide the vector by that magnitude:

\hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{(3, 6, 4)}{\sqrt{61}}

The unit vector is therefore:

\hat{v} = \left(\frac{3}{\sqrt{61}}, \frac{6}{\sqrt{61}}, \frac{4}{\sqrt{61}}\right)

Example of Unit Vector in 3D Space

Visualization of vector

v(3,6,4)

and its unit vector

\hat{v}

pointing in the same direction.

Length and Components of a Unit Vector

The defining feature of a unit vector is its length of one, which forces a relation between its components. Squaring the components and adding them gives one, and that equation governs the step from any vector to its unit vector.

Length of a Unit Vector

A unit vector has length $1$ . For this example, add the squares of the three coordinates first. A length is nonnegative, so take the positive square root at the end.

\begin{aligned} &|\hat{v}|^2 \\ &=\left(\frac{3}{\sqrt{61}}\right)^2+\left(\frac{6}{\sqrt{61}}\right)^2 \\ &\quad+\left(\frac{4}{\sqrt{61}}\right)^2 \\ &=\frac{9+36+16}{61} \\ &=\frac{61}{61} \\ &=1 \end{aligned}

Unit Vectors on Coordinate Axes

In a three-dimensional coordinate system, there are three standard unit vectors that point along the positive coordinate axes:

$\hat{\imath}$ is the unit vector in the positive $x$ -axis direction
$\hat{\jmath}$ is the unit vector in the positive $y$ -axis direction
$\hat{k}$ is the unit vector in the positive $z$ -axis direction

Every vector in $\mathbb{R}^3$ can be expressed as a linear combination of these three unit vectors.

Standard Unit Vectors

Standard unit vectors on the coordinate axes.

Recording Direction with a Unit Vector

A unit vector separates direction from magnitude, so it can describe a direction on its own. Two vectors that point the same way share one unit vector, which makes the unit vector a compact way to record a direction.

Indicating Direction

A unit vector records direction with magnitude $1$ . In physics, it can specify the direction of a force, velocity, or acceleration separately from that quantity's magnitude.

Physics Calculations

Unit vectors express a force as components along the $x$ , $y$ , and $z$ axes. Each coefficient gives the signed magnitude along one axis. A force written as $3\mathbf{i} - 4\mathbf{j}$ therefore points $3 \text{ units}$ along the $x$ -axis and $-4 \text{ units}$ along the $y$ -axis.