Magnitude and Unit Vectors (Pure Mathematics 3 – Vectors)

Core Concept

Magnitude of a Vector

The magnitude (or modulus) of a vector is its length. For a vector expressed in component form, it is found using the Pythagorean theorem extended to 3D.

For a 2D vector $\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}$, the magnitude is the straight-line distance from the origin to the point $(x, y)$.

For a 3D vector $\mathbf{a} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, an extra dimension is added. This is the single most-tested formula in the whole subtopic.

Unit Vectors

A unit vector is any vector with magnitude exactly 1. Given any non-zero vector $\mathbf{a}$, dividing by its own magnitude produces the unit vector in the same direction, denoted $\hat{\mathbf{a}}$.

Unit vectors in the directions of the coordinate axes are given special names: $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.

Position Vectors and Displacement Vectors

• A position vector of a point $P$ is the vector $\overrightarrow{OP}$ from the origin $O$ to $P$. It is conventionally written as $\mathbf{p}$, the lower-case letter matching the point label. In 3D, if $P = (a, b, c)$ then $\mathbf{p} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$.
• A displacement vector $\overrightarrow{AB}$ is the vector from point $A$ to point $B$, regardless of where $A$ lies relative to the origin. It equals $\mathbf{b} - \mathbf{a}$.

The magnitude $|\overrightarrow{AB}|$ is then the distance between $A$ and $B$.