Proving three points lie on a straight line (GCSE vectors)

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If you ever study GCSE vectors questions, you'll spot a pattern: there's normally a (relatively) straightforward first part which involves writing down a few vectors, and then something like "show that points $O$, $X$ and $Y$ lie on a straight line."

Pretty much every student I've ever worked with on this has asked 'how on earth do you do that?", so I thought I'd better reveal the deep, dark secret.

It's really easy once you know how

There are two facts you need to know:

  1. If vectors are multiples of each other, they're parallel;
  2. If two parallel vectors start at the same point, that point and the two end points are in a straight line

That means your task is easy: you just need to show that $vec{OX}$ and $vec{OY}$ are parallel1.

So, in simple steps:

  • Work out the vector $vec{OX}$;
  • Work out the vector $vec{OY}$;
  • Work out what you multiply $vec{OX}$ by to get $vec{OY}$. This may be a fraction.

Then you write down something like $vec{OX} = frac{3}{2}vec{OY}$, so $OXY$ is a straight line.

It's pretty much the same trick every time - learn it, and it'll be worth about four marks to you.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. Your letters, of course, may differ []


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