GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Vector arithmetic

A vector is a journey. However winding the route, what counts is where you started and where you finished.

Maths · Vectors

Two legs, one journey

Leg AB 11.7. Leg BC 6.7. Resultant AC 9.8. Relationship: AB + BC = AC. Slide B left or right: both legs change, but the journey still starts at A and ends at C — so the resultant AC stays exactly the same.

Leg AB11.7Leg BC6.7Resultant AC9.8Drag B left and right

AB + BC = AC. Slide B left or right: both legs change, but the journey still starts at A and ends at C — so the resultant AC stays exactly the same.

A column vector — written on one line here as (top, bottom) — is a move: top number across (positive = right, negative = left), bottom number up or down (positive = up, negative = down). In the starting position AB = (10, 6) — 10 right, 6 up — and BC = (−6, 3) — 6 left, 3 up — so the resultant AC = (4, 9). The readouts are straight-line lengths, in the same units as the vectors.

Watch out: The resultant always runs from the START (A) to the FINAL END (C). Drawn from C back to A, it points the wrong way.

Predict, then check

Multiplying a vector by a number (a scalar) changes the arrow. Commit to how, then check.

The vector a = (−3, 2) means 3 left and 2 up. What is −2a, and what does its arrow look like?

Watch the method — scale first, then combine

Problem

Work out (i) 3(2, −3) + 2(3, −1) and (ii) 2(1, 4) − 3(−2, 5).

Which would you trust?

Five students, five claims

Five students each explain one step of a vectors question in their own words.

Which ONE of these would you trust completely?

For 4(1, 2) + (3, 0), I add first to get (4, 2), then multiply by 4 to get (16, 8).(5, 1) − (2, −3) = (3, −2), because on the bottom 1 − 3 = −2.(5, 1) − (2, −3) = (3, 4), because on the bottom 1 − (−3) = 4.For a path A to B to C, I draw the resultant from C back to A.If a grid question says 'show that', drawing the resultant on its own is enough.

How sure are you?

I'm sureI think soI'm guessing

For 4(1, 2) + (3, 0), I add first to get (4, 2), then multiply by 4 to get (16, 8).

This is a common misconception

The scalar only belongs to (1, 2), and multiplications come first. 4(1, 2) = (4, 8), then (4, 8) + (3, 0) = (7, 8). Adding first multiplies the (3, 0) by 4 too — something the question never asked for.

(5, 1) − (2, −3) = (3, −2), because on the bottom 1 − 3 = −2.

This is a common misconception

The bottom sum is 1 − (−3), not 1 − 3. Subtracting a negative is adding: 1 + 3 = 4. So the answer is (3, 4). Write the bracket in — 1 − (−3) — and the sign can't escape.

(5, 1) − (2, −3) = (3, 4), because on the bottom 1 − (−3) = 4.

This matches the evidence

Exactly right. Top: 5 − 2 = 3. Bottom: 1 − (−3) = 1 + 3 = 4. You kept the whole component, sign and all, when you subtracted.

For a path A to B to C, I draw the resultant from C back to A.

This is a common misconception

The resultant is the whole journey, so it goes the way the journey went: from the start A to the final end C, arrow pointing at C. Drawn from C to A, it's the same length but the opposite direction — a different vector.

If a grid question says 'show that', drawing the resultant on its own is enough.

This is a common misconception

'Show that' means show where it comes from. Draw the original vectors, each one starting where the last one ended, and then join the start to the final end. The resultant alone doesn't show anything.

Scale first, then combine component by component — and minus a negative is a plus. Draw the resultant from the start to the final end, and when you're asked to show it, draw the vectors that make it.

Your turn to fill the gaps

Finding a missing scalar

a(3, −5) + 4(−2, 6) = (−2, 14). Find the value of a.

Scale the vector you can: 4(−2, 6) = (−8, 24). So a(3, −5) + (−8, 24) = (−2, 14).a(3, −5) is (3a, −5a) — each component times a.
missing step

Which line is step 2?

Top components: 3a − 8 = −2Top components: 3a − 8 = 14Top components: 3a + 8 = −2

Vectors with letters

Write a pathway, then mark it

Treat p and q like any letters in algebra: collect like terms, and do any multiplying before adding — just as with column vectors. A pathway is a list of moves: to get from A to C, go A to B, then B to C.

AB = 4p + q and BC = 3q. X is the midpoint of AC. (a) Write AC in terms of p and q. (b) Write AX in terms of p and q. (c) Write CB in terms of q. [4 marks]

Your answer

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The mark scheme

Writes a pathway through B: AC = AB + BC (1 mark)
AC = 4p + 4q (or the factorised form 4(p + q)) (1 mark)
AX = ½AC = 2p + 2q (1 mark)
CB = −3q (going along BC the opposite way gives its negative) (1 mark)

Model answer

(a) AC = AB + BC = (4p + q) + 3q = 4p + 4q, which is 4(p + q) in factorised form. (b) X is halfway along AC, so AX = ½AC = ½(4p + 4q) = 2p + 2q. (c) CB goes along BC backwards, so CB = −BC = −3q.

Route in, like terms together, halve for a midpoint, and flip the sign when you go the opposite way.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Scale it, add it, follow it — every vector is just a move across and a move up or down.

What you need to know

In a column vector the top number is the horizontal move (positive = right, negative = left) and the bottom number is the vertical move (positive = up, negative = down).
Multiply a vector by a scalar by multiplying each component; a negative scalar changes the sign of every component and reverses the direction.
Add or subtract column vectors component by component — but do every scalar multiplication first.
The resultant runs from the start point to the final end point, and different pathways between the same two points give the same resultant.
If one vector is a scalar multiple of another, they are parallel; going along a vector the opposite way gives its negative, and a midpoint gives half a vector.

The big picture

A column vector is a move: the top number says how far across, the bottom number how far up or down. To multiply by a scalar, multiply each component — a negative scalar flips the direction, and any scalar multiple is parallel to the original. To add or subtract, do every scalar multiplication first, then combine top with top and bottom with bottom. The resultant is the single journey from start to final end, and it is the same whichever route you take — which is how you find vectors along a pathway, even when they are written with letters.

Key points

1Top number = across, bottom number = up/down.

2Scalar × vector: multiply every component; a negative scalar flips every sign.

3Scale first, then add or subtract top with top and bottom with bottom.

4Resultant: a ruled line from the start to the final end, arrow pointing at the end.

5Scalar multiple = parallel. Opposite direction = negative. Midpoint = half.

6Same start, same finish, same resultant — whatever the route.

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Worked example

Problem

On a grid, AB = 2(5, 3) and BC = 3(−2, 1). Find the resultant AC as a column vector, and say how you would draw the diagram to show it.

⚠ Watch out

Adding the vectors before multiplying by the scalars. Do every scalar multiplication first, then add or subtract component by component.

🧠

Memory hook

Scale, stack, start-to-finish: scale each vector first, stack top with top and bottom with bottom, then draw the resultant from where you started to where you finished.

✓

Check yourself

Is (−6, 9) parallel to (−2, 3)? How can you tell — and do the two vectors point the same way or opposite ways?

Flashcards

(15)

In the column vector (a, b), what do the top and bottom numbers tell you?

Top = horizontal move (positive right, negative left). Bottom = vertical move (positive up, negative down).

What translation is the column vector (2, −5)?

2 units to the right and 5 units down.

What is displacement?

The distance from the starting point, measured in a straight line.

What is a resultant vector?

The single vector that has the same effect as a combination of other vectors.

How do you draw a resultant on a grid?

A ruled line from the start point to the final end point, with the arrowhead pointing towards the end.

How do you multiply a vector by a scalar?

Multiply each component by the scalar: 4 × (−3, 2) = (−12, 8).

What does multiplying by a negative scalar do?

Every component changes sign, so the vector points the opposite way: −1 × (−6, 10) = (6, −10).

In a sum like 3u + 2v, what do you do first?

Each scalar multiplication first — then add top with top and bottom with bottom.

How do you find a missing scalar from a known resultant?

Form an equation from one component, solve it, then check your value with the other component.

How can you tell two vectors are parallel?

One is a scalar multiple of the other — for example (4, 6) = 2 × (2, 3).

EG = 2 × EF along the same line. What does that tell you about F?

F is the midpoint of EG, so EF = ½EG.

AB = a. What is BA?

−a. Going along a vector the opposite way gives its negative.

OA = 3a − 4b and X is the midpoint of OA. What is OX?

Half of OA: 1.5a − 2b.

Two different routes go from P to Q. Do they give the same resultant?

Yes — different pathways between the same two points always give the same resultant vector.

A grid question says 'show that' the resultant is a given vector. What must you draw?

The original vectors, each starting where the last one ended — not just the resultant.

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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Five students, five claims

Finding a missing scalar

Write a pathway, then mark it

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