Unit Vectors

A unit vector is a vector with magnitude exactly $1$. It carries direction without size, which makes it useful for pointing “which way” independent of “how far”. Finding one is a short, reliable calculation built on magnitude.

Finding a unit vector

To get the unit vector in the direction of any vector $\mathbf{a}$, divide $\mathbf{a}$ by its own magnitude: $\hat{\mathbf{a}} = \frac{1}{|\mathbf{a}|}\,\mathbf{a}$

The result points the same way as $\mathbf{a}$ but has length $1$. (The “hat” $\hat{\mathbf{a}}$ conventionally denotes a unit vector.)

A worked example

Find the unit vector in the direction of $\mathbf{a} = 3\mathbf{i} - 4\mathbf{j}$. Magnitude: $|\mathbf{a}| = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5$. Unit vector: $\hat{\mathbf{a}} = \dfrac{1}{5}(3\mathbf{i} - 4\mathbf{j}) = \dfrac{3}{5}\mathbf{i} - \dfrac{4}{5}\mathbf{j}$.

Check: $\left|\hat{\mathbf{a}}\right| = \sqrt{\left(\tfrac{3}{5}\right)^2 + \left(\tfrac{4}{5}\right)^2} = \sqrt{\tfrac{9 + 16}{25}} = \sqrt{1} = 1$ ✓.

The base unit vectors i and j

$\mathbf{i}$ and $\mathbf{j}$ are themselves unit vectors, along the $x$- and $y$-axes respectively. Any vector $a\mathbf{i} + b\mathbf{j}$ is built from them, which is why writing a vector “in terms of $\mathbf{i}$ and $\mathbf{j}$” is so natural.

Why unit vectors matter

They let you scale to any length in a fixed direction: a vector of magnitude $10$ in the direction of $\mathbf{a}$ is simply $10\,\hat{\mathbf{a}}$. This appears in velocity and problem-solving questions where a speed is given along a known direction.

Common mistakes

• Dividing by $|\mathbf{a}|^2$ instead of $|\mathbf{a}|$.
• Losing the sign on a component when dividing.
• Forgetting that $\mathbf{i}$ and $\mathbf{j}$ already have magnitude $1$.

Full topic context: Vectors notes, and Vector Notation & Magnitude.