Vectors

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Cambridge IGCSE · International Maths Extended

Vectors

A vector is just an instruction for how far to move and in which direction — and once you can add, subtract, scale, and measure the length of those instructions, this whole chapter is yours.

Revision Guide · Column Vectors · Vector Diagrams · Magnitude

Summary — What This Chapter Covers

Column vectors describe a movement from one point to another — a top number for left/right, a bottom number for up/down.
You can add and subtract column vectors by combining the top numbers together and the bottom numbers together.
You can multiply a vector by a scalar (an ordinary number) — this stretches, shrinks, or flips the vector.
Vectors can be drawn as arrows on a diagram — the length shows magnitude, the arrowhead shows direction.
A vector from point to point is written AB→, and it points the way to BA→ — same length, reversed direction.
The magnitude (length) of a vector is found using Pythagoras' theorem on its two components.
Multiplying by a negative scalar reverses the direction of a vector as well as changing its length.

1. Introduction to Column Vectors

Imagine you're giving someone directions across a grid — not "turn left at the shop," but pure movement: That's exactly what a column vector is: a set of travel instructions written as a little vertical pair of numbers.

What a column vector means

( 6, 3 ) → means "6 units to the right, and 3 units up"

The top number is the horizontal (x) movement. The bottom number is the vertical (y) movement. A negative sign flips the direction — negative top = left, negative bottom = down.

Column vectors are also called translation vectors, because they describe exactly the same thing as a translation in geometry — a slide, with no rotation and no resizing. If you've done translations before, you already half-know this topic.

x ← top number → movement in x-direction (+ right, − left) Vector = ( x ) ( y ) ← bottom number → movement in y-direction (+ up, − down)

Quick way to remember it

Think "along the corridor, then up the stairs" — top number first (sideways), bottom number second (up/down). It's the same order as coordinates (x, y), just written vertically instead of with a comma.

Adding and Subtracting Column Vectors

This is the easiest operation in the whole chapter, because there's no trick — you just deal with the top numbers together, and the bottom numbers together, completely separately. Think of it like two separate number lines running in parallel: one for "sideways total," one for "up/down total."

Addition & Subtraction Rule

( a, b ) + ( c, d ) = ( a + c , b + d )

( a, b ) − ( c, d ) = ( a − c , b − d )

Add/subtract tops with tops, and bottoms with bottoms. Never mix a top number with a bottom number.

For example:

( 5 ) ( 3 ) ( 5+3 ) ( 8 ) ( 2 ) + (−1 ) = ( 2+(−1)) = ( 1 ) ( 5 ) ( 3 ) ( 5−3 ) ( 2 ) ( 2 ) − (−1 ) = ( 2−(−1)) = ( 3 )

Common mistake: students sometimes forget that subtracting a negative number means adding. In the subtraction example above, 2 − (−1) is 2 + 1 = 3, not 1. Always rewrite double negatives before calculating.

Practice Question 1

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Also in the full note

2. Representing Vectors as Diagrams
3. Magnitude of a Vector
What to Memorise
Concepts Checklist
Exam Tips

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Vectors

Vectors

Summary — What This Chapter Covers

1. Introduction to Column Vectors

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