Vector arithmetic: Addition, scalar multiplication, magnitude and unit vectors

What this dot point is asking

NESA wants you to be fluent with two-dimensional vectors: their notation (column, component and $\mathbf{i}, \mathbf{j}$ form), what they mean as arrows, how to add and subtract them both algebraically and geometrically, how to scale them, and how to find a magnitude (length) and a unit vector. This is the alphabet of the whole Vectors module. Every later dot point, the scalar product, projection, lines and geometric proofs, is written in this language, so the few minutes spent making it automatic here are repaid many times over.

The answer

A vector is an arrow: magnitude and direction

A vector carries two pieces of information at once, a magnitude (length) and a direction, and nothing else. Picture it as an arrow. Crucially, a vector is free: an arrow of the same length pointing the same way is the same vector, wherever you draw it on the page. This is why addition works by sliding arrows around, and why $\overrightarrow{AB}$ depends only on where $A$ and $B$ are, not on any axes. A quantity with size but no direction (mass, time, temperature) is a scalar, an ordinary number.

A two-dimensional vector $\mathbf{a}$ has a horizontal component $a_1$ and a vertical component $a_2$. The three notations you must read and write fluently are

where $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$ are the standard unit vectors along the axes. In handwriting a vector is underlined ($\underline{a}$) or written with an arrow; printed work uses bold. The components $a_1$ and $a_2$ tell you "how far across" and "how far up" the arrow goes, so they encode the magnitude and direction together.

Vector addition: head to tail, or the parallelogram

To add vectors, add components:

The picture is the part NESA cares about. There are two equivalent constructions, and you should be able to draw either.

• Head to tail (the triangle rule). Draw $\mathbf{a}$, then draw $\mathbf{b}$ starting from the head of $\mathbf{a}$. The sum $\mathbf{a} + \mathbf{b}$ is the single arrow from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$. It is the net displacement of "go along $\mathbf{a}$, then along $\mathbf{b}$".
• The parallelogram rule. Draw $\mathbf{a}$ and $\mathbf{b}$ from a common point. Complete the parallelogram. The diagonal from the common point is $\mathbf{a} + \mathbf{b}$.

They give the same arrow because the far side of the parallelogram is just a copy of $\mathbf{b}$ moved to the head of $\mathbf{a}$, which is exactly the head-to-tail picture.

Below, $\mathbf{a} + \mathbf{b}$ is built up one move at a time, finishing with the full parallelogram so you can see both diagonals.

Stage 1, the two vectors. Draw $\mathbf{a}$ and $\mathbf{b}$ with their tails together at a point $O$. So far nothing has been added; these are the two arrows we want to combine.

Stage 2, slide b to the head of a. A vector is free, so move a copy of $\mathbf{b}$ (keeping its length and direction) until its tail sits at the head of $\mathbf{a}$. The original $\mathbf{b}$ is left dashed for reference.

Stage 3, the sum closes the triangle. The resultant $\mathbf{a} + \mathbf{b}$ is the arrow from $O$ (the tail of $\mathbf{a}$) straight to the head of the moved $\mathbf{b}$. It closes the triangle, which is why this is called the triangle rule.

Stage 4, the parallelogram and the difference. Drawing both $\mathbf{a}$ and $\mathbf{b}$ from $O$ and completing the parallelogram gives the same diagonal $\mathbf{a} + \mathbf{b}$. The other diagonal, from the head of $\mathbf{b}$ to the head of $\mathbf{a}$, is $\mathbf{a} - \mathbf{b}$ (think "$\mathbf{b}$ plus what gets you to $\mathbf{a}$").

Vector subtraction

Subtract componentwise:

There are two reliable ways to see $\mathbf{a} - \mathbf{b}$ geometrically. Either read it as the addition $\mathbf{a} + (-\mathbf{b})$, where $-\mathbf{b}$ is $\mathbf{b}$ reversed, or read it straight off the parallelogram above: with $\mathbf{a}$ and $\mathbf{b}$ from a common point, $\mathbf{a} - \mathbf{b}$ is the arrow from the head of $\mathbf{b}$ to the head of $\mathbf{a}$. The second picture is the engine behind the displacement rule next.

Scalar multiplication: stretching and flipping

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

Geometrically $\lambda \mathbf{a}$ is parallel to $\mathbf{a}$ with magnitude $|\lambda|\,|\mathbf{a}|$: same direction if $\lambda > 0$, opposite if $\lambda < 0$, and the zero vector if $\lambda = 0$. This single idea is the definition of parallel that every proof uses: two vectors are parallel exactly when one is a scalar multiple of the other.

Magnitude by Pythagoras

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

This is just Pythagoras' theorem on the right triangle whose legs are the components $a_1$ and $a_2$ and whose hypotenuse is the arrow itself.

For a displacement $\overrightarrow{PQ} = Q - P$, the same formula gives the distance between the points:

This is literally the distance formula from coordinate geometry, which is no coincidence: the distance formula is the magnitude of a displacement vector.

Unit vectors

A unit vector has magnitude $1$. The unit vector in the direction of a non-zero $\mathbf{a}$ is

Dividing by the length rescales the arrow to length $1$ without turning it, so $|\hat{\mathbf{a}}| = 1$ by construction. Two everyday uses: to build a vector of a prescribed length $k$ in a given direction, take $k\hat{\mathbf{a}}$; and to extract a direction from a vector while throwing away its size, take $\hat{\mathbf{a}}$.

The vector laws and the triangle inequality

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 of ordinary arithmetic and let you rearrange vector expressions freely, which is exactly what geometric proofs rely on.

One law does not carry over from numbers, and it is a favourite trap. Magnitudes do not add:

with equality only when $\mathbf{a}$ and $\mathbf{b}$ point the same way. The reason is geometric: in the triangle rule, the side $\mathbf{a} + \mathbf{b}$ can never be longer than the broken path along $\mathbf{a}$ then $\mathbf{b}$, and is shorter whenever the path bends.

How exam questions ask about vector arithmetic

The wording is rarely "do vector arithmetic". Learn to translate:

• "Find $\overrightarrow{AB}$" or "the displacement from $A$ to $B$": compute $\mathbf{b} - \mathbf{a}$ (end minus start).
• "Find the distance $AB$" or "the length of $\overrightarrow{AB}$": take $|\mathbf{b} - \mathbf{a}|$.
• "Find a unit vector in the direction of ...": divide the vector by its magnitude.
• "Find the vector of magnitude $k$ in the direction of ...": form $\hat{\mathbf{a}}$, then multiply by $k$.
• "Show that $A$, $B$, $C$ are collinear": show $\overrightarrow{AB} = \lambda\,\overrightarrow{AC}$ for some scalar $\lambda$ (parallel and sharing the point $A$ forces one straight line).
• "Are $\mathbf{a}$ and $\mathbf{b}$ parallel?": test whether one is a scalar multiple of the other (equal ratios of components).
• "Express $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$": build a head-to-tail route through the named points (for example $\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = \mathbf{b} - \mathbf{a}$).