Unit Vector of a 2-D Vector

A unit vector $\hat{a}$ has magnitude $1$ and points in the direction of $\vec{a}$. The formula is $\hat{a} = \vec{a}/|\vec{a}|$. So for $\vec{a} = (3, 4)$, $|\vec{a}| = 5$ and $\hat{a} = (0.6, 0.8)$.

Enter the components of $\vec{a}$. The lab returns $|\vec{a}|$ and the unit vector $\hat{a}$. Verify $|\hat{a}| = 1$ by checking that the squared components sum to $1$. Try $\vec a = (1, 1, 1)$: $|\vec a| = \sqrt{3}$, $\hat a = (1/\sqrt 3, 1/\sqrt 3, 1/\sqrt 3)$.

They strip out magnitude and isolate direction, which is exactly what direction cosines $(l, m, n)$ are. They also simplify many formulas: angle between two lines is $\cos\theta = \hat a\cdot\hat b$. Every JEE 3-D problem starts by normalising direction vectors.

Game engines use unit vectors for surface normals to compute lighting and reflection. Robotics uses them to encode orientation. The unit vectors $\hat{i}, \hat{j}, \hat{k}$ along the axes are the building blocks of every position vector you write.