What's the Difference Between a Vector and a Scalar?

A scalar has magnitude only, while a vector has both magnitude and direction. For example, speed is scalar, but velocity is vector because it specifies how fast an object moves and in which direction.

The Key Difference

This is part of AHL 3.10: Vector definitions, so it is HL-only content in IB Mathematics: Applications and Interpretation.

A vector can be represented by its components. For example,

$$\mathbf{v}=\begin{pmatrix}3\\4\end{pmatrix}$$

means a displacement of $3$ units in the positive horizontal direction and $4$ units in the positive vertical direction. Its magnitude is

$$\lvert\mathbf{v}\rvert=\sqrt{3^2+4^2}=5.$$

Its direction, measured from the positive horizontal axis, is

$$\theta=\tan^{-1}\left(\frac{4}{3}\right)=53.1^\circ.$$

Therefore, $5$ is the scalar magnitude of the vector, while the complete vector includes both $5$ and the direction $53.1^\circ$.

Two vectors are equal only when they have the same magnitude and direction, even if they begin at different points. Vector addition combines corresponding components. This component method is central to later work with vector equations, vector kinematics, and scalar and vector products.

Multiplying a vector by a scalar changes its magnitude. For example, $2\mathbf{v}$ doubles the magnitude, while $-\mathbf{v}$ keeps the same magnitude but reverses the direction.

A common misconception is that any quantity with a negative value is a vector. This is incorrect: a scalar can be negative, such as a temperature of $-5^\circ\text{C}$, without having a spatial direction.

Exam Technique

On HL Papers 1, 2, or 3, define the distinction using the words magnitude and direction. In calculations, show the vector components before using the GDC, distinguish $\mathbf{v}$ from $\lvert\mathbf{v}\rvert$, and include the direction when the command term asks you to determine a vector rather than only its magnitude.