Perform vector arithmetic with vectors in the plane, including component and column-vector notation, and find the magnitude and unit vector

What this dot point is asking

NESA wants you to be fluent with two-dimensional vectors: their notation (column and component), addition, subtraction, scalar multiplication, magnitude (length), and the unit vector in a given direction.

The answer

Notation

A two-dimensional vector $\mathbf{a}$ has horizontal component $a_1$ and vertical component $a_2$. Common notations:

where $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$ are the standard unit vectors along the axes.

Vector addition and subtraction

Geometrically, $\mathbf{a} + \mathbf{b}$ is the diagonal of the parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$ (head-to-tail). $\mathbf{a} - \mathbf{b}$ goes from the head of $\mathbf{b}$ to the head of $\mathbf{a}$ when both start at the origin.

Scalar multiplication

For a scalar $\lambda$ and vector $\mathbf{a}$,

Geometrically, $\lambda \mathbf{a}$ is parallel to $\mathbf{a}$ (same direction if $\lambda > 0$, opposite if $\lambda < 0$) with magnitude $|\lambda| |\mathbf{a}|$.

Magnitude

The magnitude (length) of $\mathbf{a}$ is

This follows from Pythagoras' theorem on the right triangle with legs $a_1$ and $a_2$.

For a vector $\mathbf{PQ}$ from point $P = (p_1, p_2)$ to point $Q = (q_1, q_2)$,

Unit vector

The unit vector in the direction of $\mathbf{a}$ (assumed non-zero) is

By construction $|\hat{\mathbf{a}}| = 1$.

Vector laws

Vector addition is commutative ($\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$) and associative ($(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$).

Scalar multiplication distributes over addition: $\lambda (\mathbf{a} + \mathbf{b}) = \lambda \mathbf{a} + \lambda \mathbf{b}$.

These mirror the laws for real numbers and let you manipulate vector expressions algebraically.