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

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:

- If vectors are multiples of each other, they’re parallel;
- 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 parallel ((Your letters, of course, may differ)).

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.