Unit vector

Direction, stripped of everything else. Multiply a unit vector by a number and you get a vector of that size pointing that way.

AP Physics C: Mechanics EK 1.1.A.4 says vectors can be expressed in unit vector notation or as a magnitude and a direction, and EK 1.1.A.4.i adds that unit vector notation represents a vector as the sum of its constituent components in the x, y and z directions, denoted by $\hat{i}$, $\hat{j}$ and $\hat{k}$. Both Physics C equation sheets carry a VECTORS table printing $\vec{r} = (A\hat{i} + B\hat{j} + C\hat{k})$ and, for addition, $\vec{C} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.

The hat is dimensionless, so the coefficient carries the units. In $\vec{v} = 3\hat{i} - 4\hat{j}$ the numbers are metres per second and the hats supply only direction. The magnitude comes from the coefficients alone: $\lvert \vec{v} \rvert = \sqrt{3^2 + 4^2} = 5$ m/s.

Adding is why the notation exists. Components with the same hat add and components with different hats never mix, which turns vector addition into two or three separate sums with no diagram.

A hat on any symbol means the same thing. The $\hat{r}$ in the Biot-Savart law on the AP Physics C: Electricity and Magnetism sheet is a unit vector pointing from the current element toward the field point, not a distance.

Neither algebra-based course uses the notation. The phrase appears in neither framework, and neither sheet carries the VECTORS table, so those exams describe vectors by magnitude and angle or by components.