Vectors Explained: Column Notation, Adding, and Proving Lines Are Parallel

Vectors are one of the last topics most schools teach, which means they’re also the topic most students meet when they’re already tired. That’s a shame, because vectors are genuinely one of the most predictable question types on the Higher paper. The final part is nearly always “prove these two lines are parallel”, and there is a script for answering it.

Let’s build the topic from the ground up.

Start with the coordinate grid

Before vectors, remind yourself how points work. A coordinate grid has a horizontal x-axis and a vertical y-axis, crossing at the origin, written (0, 0). To read off a point, trace across to the y-axis and down to the x-axis. The point five right and three up is (5, 3); four left and two up is (−4, 2).

Remember: x always comes first. (5, 3) and (3, 5) are different points, and mixing them up is the single most common mark-loser in the early parts of a geometry question.

What a vector actually is

Here’s the idea that unlocks everything: a point has a position, but a vector does not.

A vector is drawn as an arrow, and it’s defined by only two things:

• its length (how far), and
• its direction (which way it points).

Because position isn’t part of the definition, you can pick a vector up and slide it anywhere on the page and it is still exactly the same vector. That is why the same arrow can appear three times in different corners of an exam diagram — it hasn’t changed, it’s just been moved.

We write vectors as letters with a bar on top (or in bold in printed papers, and underlined when you handwrite them).

Column vector notation

To describe a vector, count how far it travels across and how far it travels up:

• Right is positive, left is negative.
• Up is positive, down is negative.

Write those two numbers in a column, x on top and y underneath, inside square brackets. A vector going 5 right and 3 up is the column vector with 5 on top and 3 below. A vector going 4 left and 2 up has −4 on top and 2 below.

Movement	x component	y component
5 right, 3 up	5	3
4 left, 2 up	−4	2
3 right, 6 down	3	−6
2 left, 5 down	−2	−5

Tip: If you place a vector’s tail at the origin, its tip lands exactly on the point matching its components. That’s the bridge between “vector” and “coordinate” — useful when a question hands you coordinates and expects a vector answer.

Multiplying a vector by a number (scalar multiples)

When you multiply a vector by an ordinary number, that number is called a scalar — because it scales the vector. Multiply every component by it:

• Multiply by 3 → the arrow points the same way but is three times as long.
• Divide by 3 → same direction, three times shorter.
• Multiply by −1 → same length, but the arrow points the opposite way. We just call this −a.

That last one matters enormously in exam proofs. If a diagram gives you the vector from A to B as a, then the vector from B back to A is −a. Half the marks in a vector proof come from confidently reversing arrows like this.

Did you know? Two vectors are parallel exactly when one is a scalar multiple of the other. Every “prove these lines are parallel” question in the country boils down to spotting that multiple.

Adding and subtracting vectors

Adding is mechanical: add the x components, add the y components. Same for subtracting.

The geometry is what you actually need in the exam:

• To add a + b: move the tail of b to the tip of a. The answer runs from the tail of a to the tip of b — one arrow replacing two. This is the “journey” method: to get from here to there, go along a, then along b.
• To subtract a − b: the answer runs from the tip of b to the tip of a.

That subtraction rule looks arbitrary until you see why it works. If a and b both start at the origin, they behave like points, so a − b is the vector from one point to the other. Which means subtraction is how you find the route between two points — and the length of that vector is the distance between them.

Most exam diagrams are built on exactly this. You’re given a shape with vectors marked on two sides and asked for a third side; you take the route you know, reversing arrows where necessary, and add them up.

Magnitude: the length of a vector

The length of a vector is called its magnitude (or norm), and it’s written with a vertical bar on each side.

Finding it is Pythagoras in disguise. Draw a right-angled triangle underneath the arrow: the horizontal component is the base, the vertical component is the height, and the vector itself is the hypotenuse. Then:

$$\text{length} = \sqrt{x^2 + y^2}$$

So a vector of 5 across and 3 up has length √(25 + 9) = √34 = 5.83 (2 d.p.).

Notice that the signs don’t matter here — squaring removes them, so a vector going left and down has the same length as its mirror image. If you’re rusty on the triangle side of this, our trigonometry for beginners guide covers the right-angled triangle toolkit that sits underneath the whole topic.

Tip: Vectors that share a direction but not a length are still parallel. Magnitude tells you how far; the scalar multiple tells you which way. Exam questions test the two separately, so keep them separate in your head.

The exam script for a vector proof

Higher tier vector questions follow a rhythm. Learn it once and the marks become routine.

1. Label every arrow you’re given on the diagram, including the reversed versions (if AB = a, write BA = −a).
2. Find the route. To get from X to Y, travel along arrows you know — even if it’s the long way round.
3. Collect like terms. Simplify to the form ma + nb.
4. Factorise out the common number. If you get 2a + 4b, write 2(a + 2b).
5. Write the conclusion sentence. “Since XY = 2(a + 2b) and PQ = a + 2b, XY is a scalar multiple of PQ, therefore XY is parallel to PQ.” That sentence is a mark on its own — never leave it out.

Question type	What examiners want	Common slip
Write a vector in column form	Correct signs on both components	Counting up/down before left/right
Find AB from the diagram	A valid route, arrows reversed correctly	Forgetting the minus on a reversed arrow
Find the magnitude	Pythagoras, then round as asked	Rounding too early
Prove two lines are parallel	Factorising, plus the conclusion sentence	Doing the algebra but never writing “parallel”
Prove a point is a midpoint	Showing one vector is half the other	Stopping at the algebra

Where vectors sit in the wider Geometry paper

Vectors don’t live alone. They’re stitched into the same part of the paper as circle theorems and bearings — the “reason your way to an answer” questions that reward clear written justification over frantic calculation. If you can already write a decent circle theorem reason, you have the exact skill a vector proof needs.

The overlap with algebra is real too. Collecting like terms in 3a + 2b − a + 4b is ordinary algebra wearing a geometry costume, so if that step feels wobbly, spend twenty minutes on the algebra fundamentals guide before you touch another vector question.

Remember: in a vector proof, the algebra earns the method marks but the sentence earns the final one. Examiners are marking your reasoning, not just your arithmetic.

Practise vectors until the routes are automatic

Vectors reward repetition more than almost any other topic, because the difficulty isn’t the maths — it’s spotting the route through the diagram quickly. That comes from volume.

GCSE Geometry is where to drill this: visual practice across 2D & 3D shapes, angles, trigonometry, transformations and circle theorems, so you’re working vectors alongside the topics they get combined with in real papers. Everything is diagram-led, which matters for a topic you have to see before you can solve.

If you’d rather revise everything in one place, GCSE Maths bundles Number, Algebra, Geometry and Statistics together — 2900+ questions across the four subject apps — so you can switch from vector proofs to simultaneous equations without juggling four separate installs.

Start with column notation until it’s automatic, move to routes around a diagram, and finish with full proofs. Do that for a fortnight and vectors stop being the scary last question — they become the one you’re pleased to see.