Skip to content
Edexcel Pure Paper 2 2024 Question 7
Edexcel Pure Paper 2 2024 Question 7 – Vectors and Position Vectors
❓ The Question
🧠 Before you start
This is one of those vector questions that doesn’t look too bad… but if you rush it, it can go off track quite quickly.
There are only a couple of steps, really. But they depend on each other, so if the first one isn’t right, everything after it feels harder than it should.
Best approach here is just to keep things tidy. Write out the vectors clearly and don’t try to shortcut anything.
✏️ Working
We’re given two position vectors:
\vec{OA} = 2\mathbf{i} – 3\mathbf{j} + 5\mathbf{k}
\vec{OB} = 5\mathbf{i} + 6\mathbf{j} + 8\mathbf{k}
Part (a)
We’re asked to find \vec{AB}.
The standard approach here is:
\vec{AB} = \vec{OB} – \vec{OA}
So just subtract component by component.
For the \mathbf{i} terms:
5 – 2 = 3
For \mathbf{j}:
6 – (-3) = 9
For \mathbf{k}:
8 – 5 = 3
So:
\vec{AB} = 3\mathbf{i} + 9\mathbf{j} + 3\mathbf{k}
Nothing complicated here — just careful subtraction.
Part (b)
Now this is where it changes slightly.
We’re told that point P lies on the line AB, and there’s a relationship between the lengths of \vec{AP} and \vec{BP}.
That’s the key idea — we’re working with a ratio along the line.
Instead of jumping straight into coordinates, it helps to think about what this actually means.
If P lies on AB, then its position vector can be written as:
\vec{OP} = \vec{OA} + \lambda \vec{AB}
So substitute what we already have:
\vec{OP} = (2\mathbf{i} – 3\mathbf{j} + 5\mathbf{k}) + \lambda (3\mathbf{i} + 9\mathbf{j} + 3\mathbf{k})
If you expand that:
\vec{OP} = (2 + 3\lambda)\mathbf{i} + (-3 + 9\lambda)\mathbf{j} + (5 + 3\lambda)\mathbf{k}
Now we use the condition involving the lengths.
The question tells us there’s a relationship between |\vec{AP}| and |\vec{BP}|.
That gives us an equation involving \lambda.
At this stage, it becomes algebra. You substitute expressions for AP and BP in terms of \lambda, then simplify.
It’s not especially quick, but it’s manageable if you keep everything organised.
Solving that equation gives you the possible values of \lambda.
Final step
Once you have \lambda, you substitute it back into the expression for \vec{OP}.
That gives the position vectors of the point P.
You’ll usually end up with two possible answers, depending on the values of \lambda.
🎯 Where the Marks Are
Part (a) is just one step — straightforward vector subtraction.
Part (b) carries most of the marks. Not because it’s conceptually difficult, but because there are several stages:
- setting up the vector equation
- using the condition properly
- solving for \lambda
- substituting back
Each of those is worth something.
⚠️ What Went Wrong
A few common issues showed up here.
Some answers mixed up the direction of \vec{AB}, which throws everything off from the start.
Others tried to jump straight to coordinates without using a parameter like \lambda, which made things more complicated than necessary.
There were also algebra slips when forming the equation from the length condition — especially with squares and roots.
💡 One Small Tip
If you see a point on a line between two vectors, think “parameter”.
Writing it in terms of \lambda usually makes the rest much clearer.
🚀 If This Felt Difficult
If this felt a bit heavy, it’s probably because there’s a mix of ideas — vectors, algebra, and interpreting a condition.
That’s quite common in exam questions.
Working through more of these helps you boost your maths confidence, especially when you start recognising the setup straight away.
And if you want something more structured, a step-by-step maths revision course can help build that familiarity so questions like this don’t feel as long.
🔗 Next Steps
👨🏫Author Bio
S Mahandru is an A Level Maths teacher focused on helping students stay accurate under exam pressure. The aim is to keep methods clear and build confidence through consistent exam-style practice.
❓ Frequently Asked Questions
📌Why use OB − OA?
Because that gives the vector from A to B directly.
📌Why introduce λ?
It allows you to describe any point along the line AB.
📌Do I always get two answers?
Often, yes — depending on the equation you solve.
📌Is this a standard exam question?
Yes, vector questions like this appear regularly.