Vectors is one of the most predictable topics in the O-Level E Math syllabus (4048), and yet it is where a lot of students quietly lose four or five marks every year. The maths itself is not hard. What trips people up is notation, direction, and not knowing the exact sentence a marker wants to see.
Here is how we teach it at Dojo, in the order you should actually attack the question.
What Paper 2 vectors questions usually look like
In Paper 2 you will typically get one structured question worth around 6 to 9 marks, built on a figure with a triangle or a parallelogram. The parts almost always follow this shape:
- Express one or two vectors in terms of a and b.
- Express a longer vector using a given ratio.
- Show that two points and a third are collinear, or find a ratio such as AP : PB.
- Find a ratio of areas.
Parts (a) and (b) feed into (c) and (d). If you get the first expression wrong, the rest collapses. So slow down at the start.
Step 1: Fix your notation before you write anything
Markers do check this. In handwriting, vectors must be underlined, for example a, or written with the arrow notation for AB. Writing plain "AB" without the arrow can cost you, because AB on its own means a length.
Also remember:
- BA is the negative of AB. Reversing the letters flips the sign.
- A position vector is always measured from the origin O, so OP is the position vector of P.
- The magnitude of a column vector with components x and y is the square root of x squared plus y squared.
Step 2: Use the route method
Every vector question is a journey. To get from A to B, take any path made of vectors you already know.
AB = AO + OB = negative a plus b, which is b minus a.
Say it out loud as you write: "back to O, then out to B." The letters must chain, so the end letter of one vector matches the start letter of the next. If they do not chain, you have made a sign error.
Step 3: Handle ratios on a line correctly
Suppose OA = a, OB = b, and P lies on AB with AP : PB = 1 : 2.
Start at O, walk to A, then walk one third of the way along AB:
OP = OA + (1/3)AB = a + (1/3)(b minus a) = (2/3)a + (1/3)b.
The common mistake is using 1/2 because the ratio says 1 : 2. The fraction of the whole segment is 1 out of 3 parts, not 1 out of 2. Always convert the ratio into parts of the total first.
Step 4: Proving parallel and collinear
This is where the marks are won or lost, because it is a communication mark as much as a working mark.
If OQ = 2a + b, notice that
OQ = 3 times ((2/3)a + (1/3)b) = 3 OP.
Now write the full conclusion:
"Since OQ = 3 OP, OQ is parallel to OP. As they share the common point O, the points O, P and Q are collinear."
The three ingredients the mark scheme wants are: one vector expressed as a scalar multiple of the other, the word parallel, and the common point. Miss the common point and you have only proved parallel, not collinear.
You can then read off OP : PQ = 1 : 2 directly from the scalar 3.
Step 5: Area ratios
Two rules cover almost every case.
- Same height: two triangles sharing the same apex and with bases on the same straight line have areas in the ratio of their bases. So area of triangle OAP : area of triangle OAB = AP : AB = 1 : 3.
- Similar triangles: if lengths are in ratio k, areas are in ratio k squared.
Write the reason next to the answer. Something like "same perpendicular height from O to line AB" is enough and often carries a mark.
Common mistakes to check for
- Leaving the answer as a ratio in the wrong order. If the question asks for AP : PB, do not give PB : AP.
- Forgetting to simplify to the form ma + nb with single fractions.
- Mixing up magnitude and vector. A magnitude is a number and has no direction.
- Using a diagram measurement instead of algebra. Figures in Paper 2 are not drawn to scale.
How to practise this efficiently
Do not do twenty different vector questions. Do the same five past-paper structured questions three times each until the route method and the collinearity sentence come out automatically. Ten Year Series questions from the last five years are more than enough coverage for this topic.
If you want a tutor who sat these exact papers recently to walk you through a full vectors question line by line, message us and start a free trial class on WhatsApp: https://wa.link/dsgbkf