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:

  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 []


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

  • manisha

    Your tutoring really helped me-thanks!… and my exams are almost! But, one thing I’m confused is about proving whether it is a straight line using vector method?

    • Colin

      Thanks, Manisha – where do you get lost if you use the method in this post? (If you let me know where it gets confusing, I can try to fix it for you!)

  • cecilia

    hi, do you have to have it so that the vector on the right side of the equal sign is the one with the nunber in front, as i did a vector question nd i didnt know whether to write 2NM = MC or NM = 1/2MC? thank you

  • cecilia

    hi, do you have to have it so that the vector on the right side of the equal sign is the one with the nunber in front, as i did a vector question and i didnt know whether to write 2NM = MC or NM = 1/2MC? thank you

    • Colin

      Either way is absolutely fine! I’d probably go for $2NM = MC$, myself, but that’s a point of style rather than correctness.

  • dina

    I used to overcomplicate vector questions like these, but splitting the straight line into 2 sections with the same starting point just seems so obvious! Thank you for this method, I answered a ‘Prove..’ question correctly for the first time 🙂

    • Colin

      ROCK ON! Really pleased to hear that 🙂

  • Minuka

    I took the example:-
    O,A and B are three points such that OA vector = a OB vector = b , mark the points C,D,E in a diagram such that OC vector = a+b , OD vector = 1/2a +b and OE=1/3b,
    Given that F is the mid point of OD, Show that E,F,C lie on a straight line

    If anyone do this example right they will fully understand the conditions for 3 more more vectors to lie on the same line

  • Tina

    Hey just discovered your website…
    I’ve given the x movement and y movement of 2 of the vectors in that bracket () form and they are same. So is it one of the right methods to prove that they are in a straight line?

    • Colin

      That should work, yes!

  • A pleased individual

    Dude. Dude. This is a perfect explanation of vectors. I’ve honestly been trying to find a solution to vectors that makes ACTUAL SENSE for ages. So thank you (: Very helpful

  • Prathibha Chandrashekar

    can u please explain in geometrical method that three points lie on a same line.

    • Colin

      What information are you given? A typical method might involve similar triangles, but without knowing what question you’re working on, I can’t really tell you!

  • Akhil Evilprince

    I have a question:
    The position vectors of A, B and C, relative to an origin are 2i+4j , 6i+8j and ki+25j , where k is positive.
    Find the value of k for which ABC is a straight line.

    Please help me solve this problem

    • Colin

      What have you tried?

      Hint: if A, B and C are in a straight line, then \vec{AB} = \lambda \vec{AC}$.

      • Edgar


  • Louise Hudson

    Thankyou so much this really helped me with vectors!

  • Sam

    Thanks 🙂 I finally learnt how to do this properly now! I always used to forget but now I know this: “If vectors are multiples of each other, they’re parallel;” I remember it

    • Colin

      Awesome, Sam!

  • Farha

    Thank you so much for such an easy explaination. Its really very helpful .

  • Freddie

    I’m stuck, how do you find the vector of a kind, I haven’t been given coordinates?? just that O to A is 2a?? and O to B is 2b, how could I work this out??

    • Colin

      I’m sorry, I don’t understand your question.

  • Mia Ford

    I’ve got up to 2a-b:6a-8b and I’m not sure how to see what I multiply to get the answer. Please help me

    • Colin

      I would start by factorising out as much as I could: the first of those doesn’t have any non-trivial common factors, but the second is $2(3a-4b)$.

      That’s not a multiple of the first, so the two vectors you have there are not parallel.

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