GCSE · Maths · AQA · Spec 8300 · Foundation

Vector addition, subtraction, scalar multiplication, column vectors

Ask for directions and you'll hear "three streets along, then two up". That's a vector: a movement with a size and a direction. Now let's add, subtract and stretch them.

Vectors · Adding arrows

Walk a, then walk b

a across 4.0. b across 2.0. a + b across 6.0. a up 1.0. b up 3.0. a + b up 4.0. Classification: tip-to-tail. Relationship: Across: a's run + b's run = the run of a + b. (When b points back to the left, take its run away instead.) Up: 1 + 3 = 4.

a across4.0b across2.0a + b across6.0a up1.0b up3.0a + b up4.0drag R left and right

tip-to-tail

Across: a's run + b's run = the run of a + b. (When b points back to the left, take its run away instead.) Up: 1 + 3 = 4.

Arrow a goes from P to Q. Arrow b starts where a stops and goes from Q to R. The bold line straight from P to R is a + b. Every run is measured in grid squares.

Predict, then check

A column vector stacks two numbers: the top one is the movement left or right, the bottom one is the movement up or down. Here we write it as (top over bottom).

Mo walks from S to F along the grid lines: 5 squares left, then 2 squares up, then 2 squares right. Which column vector takes you straight from S to F?

Subtracting, two ways

Problem

a = (5 over 2) and b = (1 over −3). Work out a − b, once with the columns and once on the grid, and check both give the same arrow.

Multiplying by a number

What does a number do to a vector?

a = (3 over −2): 3 right and 2 down. Now think about 2a and −a. You've already met one of these, because turning b round in the subtraction made −b.

Which is closest to what you think right now?

Only the top number changes, so −a = (−3 over −2).−a is a smaller arrow. Multiplying by a negative shrinks it.The number multiplies both entries: 2a = (6 over −4) and −a = (−3 over 2).You can multiply two vectors the same way: top times top, bottom times bottom.

How sure are you?

I'm sureI think soI'm guessing

Only the top number changes, so −a = (−3 over −2).

This is a common misconception

The number multiplies every entry in the column, not just the first one. −1 × −2 = 2, so −a = (−3 over 2). If you only flip the top, you get an arrow pointing somewhere new, not a pointing back the other way.

−a is a smaller arrow. Multiplying by a negative shrinks it.

This is a common misconception

Tempting, because negative numbers feel smaller. But work it out: −a = (−3 over 2), which is 3 left and 2 up. Those are the same distances as a, so the arrow is exactly as long as a. The minus sign changes the direction, not the size.

The number multiplies both entries: 2a = (6 over −4) and −a = (−3 over 2).

This matches the evidence

Spot on. A number used like this is called a scalar, and it scales the whole arrow. 2a points the same way as a and is twice as long. −a is exactly as long as a and points the opposite way. Put the two together and −2a is twice as long and reversed.

You can multiply two vectors the same way: top times top, bottom times bottom.

This is a common misconception

That looks like the rule for adding, but there's no such move here. A vector can be multiplied by a scalar (an ordinary number like 2 or −1), but in this topic two vectors are never multiplied by each other.

A scalar multiplies both numbers in the column. 2a, 3a and so on point the same way as a and are 2, 3… times as long. Multiplying by −1 reverses the direction and keeps the length.

Your turn to fill the gaps

Scale, then subtract

a = (3 over 2) and b = (1 over 4). Work out 2a − 3b.

2a − 3b: scale each vector first, then subtract.Two jobs, in that order.
missing step

Which line is step 2?

2a = (6 over 2)2a = (6 over 4)2a = (5 over 4)

Spot the slip

Where does this answer go wrong?

p = (2 over −3) and q = (4 over −5). Work out p − q.

A student's answer — which line goes wrong?

p − q = (2 − 4 over −3 − (−5))
Top: 2 − 4 = −2
Bottom: −3 − (−5) = −8
p − q = (−2 over −8)
So p − q moves 2 left and 8 down.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

a + b: walk a, then walk b

A vector is one journey written two ways: an arrow on a grid, or a column of two signed numbers. Adding, subtracting and scaling give the same answer whichever way you write it.

What you need to know

A column vector has two numbers: the top one is the movement right (+) or left (−), the bottom one is the movement up (+) or down (−).
Add or subtract vectors top with top and bottom with bottom, or draw them tip-to-tail.
A scalar multiplies both numbers in the column; −1 reverses the direction and keeps the length.
Vectors are multiplied by scalars, never by each other, in this topic.

The big picture

A vector is a movement with a size and a direction. You can draw it as a straight arrow from start to end, or write it as a column vector: the top number is the movement left or right, the bottom number is the movement up or down. Add or subtract vectors by combining top with top and bottom with bottom, or by drawing them tip-to-tail. Multiplying by a scalar multiplies both numbers; multiplying by −1 reverses the direction and keeps the length.

Key points

1A vector is the straight line from its start point to its end point (the shortest route), and it has both a size and a direction.

2Give the direction as well as the distance: "3 left", not just "across 3".

3a + b: walk a, then start b where a ends. The resultant is the straight arrow from the very start to the very end, and its column is the sum of the two columns.

4a − b = a + (−b): in columns, subtract top from top and bottom from bottom; on the grid, reverse b and add it on.

5ka multiplies both numbers by k. 2a points the same way as a and is twice as long; −a is the same length as a, pointing the opposite way.

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Worked example

Problem

u = (−2 over 5) and v = (3 over −1). Work out u + 2v, then describe the movement in words.

⚠ Watch out

Multiplying only the top number by the scalar. 3 × (2 over −1) is (6 over −3), not (6 over −1): the scalar multiplies both numbers.

🧠

Memory hook

Along the corridor, then up or down the stairs: the top number goes along, the bottom number goes up or down.

✓

Check yourself

Draw a = (2 over 1), then 3a and −a from the same start. Does 3a point the same way and reach three times as far? Is −a exactly as long as a, pointing back?

Flashcards

(11)

In a column vector, what do the top and bottom numbers mean?

Top: the movement right (+) or left (−). Bottom: the movement up (+) or down (−).

What does a vector look like when it is drawn?

A straight arrow from the start point to the end point. It is the shortest route, and it has a size and a direction.

Why is "across 3" not a full description of a vector?

A vector has a direction as well as a size, so you must say which way: 3 right or 3 left.

How do you add two column vectors?

Add the top numbers together and add the bottom numbers together.

How do you add two vectors by drawing?

Tip-to-tail: start b where a ends. a + b is the straight arrow from the start of a to the end of b.

How do you subtract column vectors?

Top minus top, bottom minus bottom, keeping the order. Taking away a negative number means adding.

How can you draw a − b?

Reverse b so it points the other way, then join it on to the end of a: a − b = a + (−b).

What is a scalar, and what does it do to a column vector?

An ordinary number, like 2 or −1. It multiplies both numbers in the column.

How does the arrow for 3a compare with the arrow for a?

It points the same way and is three times as long.

How does −a compare with a?

Same length, opposite direction. Multiplying by −1 changes the direction only.

Can you multiply two vectors together in this topic?

No. A vector can be multiplied by a scalar, but vectors are not multiplied by each other.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What does a number do to a vector?

Scale, then subtract

Where does this answer go wrong?

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