1.2.5. Unit Vectors

Table of Contents

Direction as a Quantity

A vector has both magnitude and direction. Sometimes we want to describe direction by itself, without changing the size of the vector. A unit vector does exactly this. It is a vector whose magnitude is 1, and it points in a specific direction.

If a vector tells us both "how much" and "which way", then a unit vector tells us only "which way". This makes unit vectors very useful in physics, because we can separate a vector into its size and its direction.

For example, if a force has magnitude $10 \, \text{N}$ and points to the right, we can write it as magnitude times direction. The direction is represented by a unit vector.

What a Unit Vector Is

A unit vector is any vector with magnitude equal to 1. If we call a unit vector $\hat{u}$, then its magnitude is

$$
|\hat{u}| = 1
$$

The hat symbol is commonly used to show that a vector is a unit vector.

A unit vector has magnitude 1. Its main purpose is to represent direction only.

If a vector $\vec{A}$ has magnitude $A = |\vec{A}|$, then we can write

$$
\vec{A} = A \hat{A}
$$

where $\hat{A}$ is the unit vector in the direction of $\vec{A}$.

This means that every nonzero vector can be thought of as

$$
\text{vector} = \text{magnitude} \times \text{direction}
$$

Finding the Unit Vector of a Given Vector

Suppose we already know a vector $\vec{A}$. To find the unit vector that points in the same direction, we divide the vector by its magnitude:

$$
\hat{A} = \frac{\vec{A}}{|\vec{A}|}
$$

This works only if $\vec{A} \ne \vec{0}$, because the zero vector has magnitude zero, and division by zero is not defined.

For any nonzero vector $\vec{A}$, the unit vector in its direction is
$$
\hat{A} = \frac{\vec{A}}{|\vec{A}|}
$$
This cannot be done for the zero vector.

As a simple example, let

$$
\vec{A} = 3\hat{i} + 4\hat{j}
$$

Its magnitude is

$$
|\vec{A}| = \sqrt{3^2 + 4^2} = 5
$$

So the unit vector in the direction of $\vec{A}$ is

$$
\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}
$$

Its magnitude is 1, as expected.

Standard Unit Vectors in Cartesian Coordinates

In Cartesian coordinates, the most common unit vectors point along the coordinate axes. In two and three dimensions, these are written as $\hat{i}$, $\hat{j}$, and $\hat{k}$.

The vector $\hat{i}$ points in the positive $x$ direction, $\hat{j}$ points in the positive $y$ direction, and $\hat{k}$ points in the positive $z$ direction.

Their magnitudes are all equal to 1:

$$
|\hat{i}| = |\hat{j}| = |\hat{k}| = 1
$$

These vectors are the basic directional building blocks for many vectors in physics.

Unit vector	Direction
$\hat{i}$	Positive $x$ axis
$\hat{j}$	Positive $y$ axis
$\hat{k}$	Positive $z$ axis

Writing Vectors with Unit Vectors

A vector can be written as a combination of standard unit vectors. In three dimensions,

$$
\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}
$$

Here, $A_x$, $A_y$, and $A_z$ are numbers that tell us how much of the vector lies in each axis direction.

This notation is compact and very useful. It clearly shows both the components and the directions.

For example,

$$
\vec{F} = 5\hat{i} - 2\hat{j}
$$

means the vector has a component of $5$ in the positive $x$ direction and a component of $2$ in the negative $y$ direction.

Magnitude from Component Form

When a vector is written using unit vectors, its magnitude can be found from its components. In two dimensions,

$$
|\vec{A}| = \sqrt{A_x^2 + A_y^2}
$$

In three dimensions,

$$
|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
$$

Then the corresponding unit vector is

$$
\hat{A} = \frac{A_x \hat{i} + A_y \hat{j} + A_z \hat{k}}{\sqrt{A_x^2 + A_y^2 + A_z^2}}
$$

If
$$
\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}
$$
then
$$
\hat{A} = \frac{\vec{A}}{|\vec{A}|}
$$
with
$$
|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
$$
for a nonzero vector.

Geometric Meaning

Geometrically, unit vectors act like arrows of length 1. Their job is to mark direction on a coordinate system. When multiplied by a number, they create vectors of different magnitudes in the same direction.

For instance,

$$
7\hat{i}
$$

is a vector of length 7 pointing along the positive $x$ axis, while

$$
-3\hat{j}
$$

is a vector of length 3 pointing along the negative $y$ axis.

So the number controls the size, and the unit vector controls the direction.

Rendering scientific sketch…

Standard Cartesian unit vectors

Unit Vector from Two Points

Sometimes we want the unit vector that points from one point to another. Suppose point $P_1$ has position vector $\vec{r}_1$ and point $P_2$ has position vector $\vec{r}_2$. The vector from $P_1$ to $P_2$ is

$$
\vec{r}_{12} = \vec{r}_2 - \vec{r}_1
$$

The unit vector in that direction is

$$
\hat{r}_{12} = \frac{\vec{r}_2 - \vec{r}_1}{|\vec{r}_2 - \vec{r}_1|}
$$

This idea is very important in physics when describing directions between objects, such as forces or motion along a line between two points.

Common Examples in Physics

In physics, unit vectors are used whenever direction matters. A displacement, velocity, acceleration, force, or electric field can all be written as magnitude times a unit vector.

For example, a force of magnitude $12 \, \text{N}$ upward could be written as

$$
\vec{F} = 12\hat{j}
$$

A velocity of magnitude $20 \, \text{m/s}$ in the direction of a unit vector $\hat{u}$ is

$$
\vec{v} = 20\hat{u}
$$

This form is especially useful because it separates physical size from physical direction.

Important Distinction

A component such as $A_x$ is not itself a vector. It is just a number. The actual vector contribution along the $x$ direction is

$$
A_x \hat{i}
$$

This distinction matters. The quantity $3$ is not the same as $3\hat{i}$. The first is only a scalar number, while the second is a vector pointing in the positive $x$ direction.

Do not confuse a component with a vector.
$$
A_x \ne A_x \hat{i}
$$
The component is a scalar. The term with the unit vector is a vector.

Summary Table

Idea	Expression
Magnitude of a unit vector	$1$
Unit vector from a nonzero vector $\vec{A}$	$\hat{A} = \dfrac{\vec{A}}{	\vec{A}	}$
Vector written as magnitude times direction	$\vec{A} =	\vec{A}	\hat{A}$
Standard unit vectors	$\hat{i}, \hat{j}, \hat{k}$
3D vector form	$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

Final Idea

Unit vectors are the simplest vectors possible, because they carry direction without extra size. They let us write any nonzero vector in a clean and organized way, especially in coordinate form. Once you understand unit vectors, writing and interpreting vectors becomes much easier.

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