GCSE · Maths · AQA · Spec 8300 · Higher

Vector geometric arguments and proofs (Higher)

Three points look as if they line up. Looking isn't proving, but vectors can make it certain in a few lines of algebra.

Vectors · geometric proof

Slide Q until M, P and Q line up

angle MPQ 155°. MP 2.0. PQ 2.7. MN 3.9. AB 7.8. Relationship: Line them up and read the lengths: angle MPQ = 180° and PQ = 2 × MP. The two segments point the same way AND meet at P.

angle MPQ155°MP2.0PQ2.7MN3.9AB7.8drag Q along the line

Line them up and read the lengths: angle MPQ = 180° and PQ = 2 × MP. The two segments point the same way AND meet at P.

OAB is a triangle. M is the midpoint of OA, P is on AB with AP : PB = 2 : 1, and Q slides along the line through O and B. Drag Q and watch the angle at P. When it reads 180°, the path M → P → Q is one straight line.

Watch out: Now look at MN and AB. They point the same way and MN is exactly half as long as AB, but they never meet. Pointing the same way is not enough to put points on one line.

Maths · Algebra

Now prove it: the line you just found

Same triangle as the board, with Q where it lined up: exactly as far past B as O is before it. Step through; each line says what it does in the argument.

GoalOA = a, OB = b. M is the midpoint of OA, AP : PB = 2 : 1, and B is the midpoint of OQ. Prove that M, P and Q are collinear.

AB = AO + OB = −a + b

route

Start with the side P sits on. A to O runs against the arrow of a, so that leg is −a.

Step 1 of 7

Start with the side P sits on. A to O runs against the arrow of a, so that leg is −a.

Watch out: AB is −a + b, not a − b. Start at A, go back to O (against a), then out along b.

Check your thinking

What did PQ = 2MP actually prove?

You've just shown that PQ = 2MP, where M, P and Q are three points.

Which is closest to what you think right now?

PQ is parallel to MP. They also both pass through P, so M, P and Q are on one straight line.PQ = 2MP already proves M, P and Q are collinear. There's nothing more to add.The diagram already shows the points lining up, so the algebra is just a check.They can't be parallel. PQ is twice as long as MP, and parallel vectors must be the same length.It works because the 2 is positive. If it were PQ = −2MP, they wouldn't be parallel.

How sure are you?

I'm sureI think soI'm guessing

PQ is parallel to MP. They also both pass through P, so M, P and Q are on one straight line.

This matches the evidence

Exactly. The multiple gives you 'parallel'. The shared point P pins the two segments onto the same line. A full proof says both, in that order.

PQ = 2MP already proves M, P and Q are collinear. There's nothing more to add.

This is a common misconception

Nearly, but the multiple only proves the two vectors point the same way. Think of MN and AB on the board: MN is half of AB and parallel to it, yet they sit on two separate lines. What rescues PQ and MP is that they share the point P. Without saying so, the proof is unfinished.

The diagram already shows the points lining up, so the algebra is just a check.

This is a common misconception

A diagram can fool your eye. Drag Q one step either side of the right spot and the angle reads 178°, which looks dead straight but isn't. Only the algebra shows it has to be exactly straight.

They can't be parallel. PQ is twice as long as MP, and parallel vectors must be the same length.

This is a common misconception

Parallel is about direction, not length. Multiplying a vector by 2 doubles its length but keeps its direction. That's exactly why a scalar multiple proves parallel.

It works because the 2 is positive. If it were PQ = −2MP, they wouldn't be parallel.

This is a common misconception

A negative multiple still means parallel. −2MP is twice as long as MP and points the opposite way, but the two vectors are still parallel.

A scalar multiple proves parallel. Parallel plus a shared point proves collinear. Write both.

Your turn

Write the whole proof yourself

OAB is a triangle with OA = a and OB = b. X is the point on AB with AX : XB = 1 : 3. Y is the point with BY = 3a. Prove that O, X and Y are collinear. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

AB = −a + b (or b − a) (1 mark)
OX = OA + ¼AB = ¾a + ¼b (1 mark)
OY = OB + BY = 3a + b, and this is 4 × (¾a + ¼b), so OY = 4OX (1 mark)
Conclusion: OY is a multiple of OX, so they are parallel, and they share the point O, so O, X and Y are collinear (1 mark)

Model answer

AB = AO + OB = −a + b. AX is 1 of the 4 equal parts of AB, so AX = ¼(−a + b). OX = OA + AX = a + ¼(−a + b) = ¾a + ¼b = ¼(3a + b). OY = OB + BY = b + 3a = 3a + b. So OY = 4 × ¼(3a + b) = 4OX. OY is a scalar multiple of OX, so OY is parallel to OX. They share the point O, so O, X and Y are collinear.

Here the shared point is O, not a point in the middle. Any point that both vectors pass through will do, as long as you name it.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A vector proof turns 'it looks straight' into 'it has to be straight'.

What you need to know

Treat every vector as a journey. To get from one point to another, build a route through points you already know and add up the legs.
Going against the direction of a given vector flips its sign: if OA = a, then AO = −a.
If P splits AB in the ratio m : n, then AP is m/(m + n) of AB. For AP : PB = 2 : 1, AP = ⅔AB.
If one vector is a scalar multiple of another, like PQ = 2MP, the two vectors are parallel.
Parallel vectors that also share a point lie on one straight line, so the points are collinear. To prove collinear, you must show both.

The big picture

To prove points are collinear, write the vectors between them as routes through known vectors, simplify and factorise to show one is a scalar multiple of the other (so they are parallel), then name the point they share.

Key points

1Choose routes that only pass through points you can already describe with the given vectors. The origin O is often the easiest stop.

2Simplify each route vector fully, then factorise so the bracket is as simple as possible. Matching brackets reveal the scalar multiple.

3A negative multiple still means parallel: PQ = −3RS means PQ is parallel to RS but points the opposite way.

4The multiple also tells you the length ratio: PQ = 2MP means PQ is twice as long as MP, so MP : PQ = 1 : 2.

5It works backwards too. If three points are collinear, the vector between one pair is a multiple of the vector between another pair, which lets you find an unknown.

6Finish with a full sentence: '… is a multiple of …, so they are parallel. They share the point …, so … are collinear.'

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Worked example

Problem

OABC is a parallelogram with OA = a and OC = c. M is the midpoint of AB and N is the midpoint of BC. Prove that MN is parallel to AC.

⚠ Watch out

Writing AB = a − b when OA = a and OB = b. From A you go back along a (−a), then out along b, so AB = −a + b. Get the sign wrong and the brackets won't match.

🧠

Memory hook

Multiple means parallel. Parallel plus a meeting point means one straight line.

✓

Check yourself

Without looking back: what does a scalar multiple prove about two vectors, what extra fact turns that into 'the points are collinear', and if OA = a, what is AO?

Flashcards

(7)

u = 3v. What does that tell you about u and v?

They are parallel: same direction, and u is 3 times as long as v.

What two facts prove that points X, Y and Z are collinear?

A scalar multiple, e.g. XY = kYZ, so the vectors are parallel, AND a shared point (here Y) joining them.

OA = a. What is AO?

−a. Going against the arrow flips the sign.

P is on AB with AP : PB = 2 : 3. What fraction of AB is AP?

⅖. Add the parts (2 + 3 = 5) to get the denominator; AP takes 2 of them.

How do you find a vector between two points you can't join directly?

Go via points you know and add the legs, e.g. MQ = MO + OQ.

Why factorise a route vector such as 2a − 8b?

2(a − 4b) shows the bracket it shares with other vectors, so a scalar multiple is easy to spot.

PQ = −3RS. Parallel or not?

Parallel. The minus sign means PQ points the opposite way, 3 times as long.

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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Now prove it: the line you just found

What did PQ = 2MP actually prove?

Write the whole proof yourself

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