Show WorkingWorked example 1: adding and multiplying column vectors
a = \vec−6−5 and b = \vec−2−2
Work out a + b as a column vector.
- Add the top numbers, then the bottom numbers: top −6 + (−2) = −8, bottom −5 + (−2) = −7.
- a + b = \vec−8−7
Answer \vec−8−7
Geometry and measuresFoundation and Higher (some Higher only)
Vectors GCSE questions with answers
8 vectors questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.
Vector questions ask you to add, subtract and multiply column vectors. On Higher you also write vectors in shapes using a and b, including routes through midpoints and points that split a line in a given ratio.
01 / Quiz yourselfTry it, then tap to check
Vectors questions to try
Work each one out on paper first. Then open the answer and compare your working with ours, step by step.
01 / Adding and multiplying column vectorsBoth tiers
a = \vec−5−1 and b = \vec−16
Work out 2a − 4b as a column vector.
Show the answer and working
- Multiply both numbers in a by 2: 2a = \vec−10−2
- Multiply both numbers in b by 4: 4b = \vec−424
- Subtract the top numbers, then the bottom numbers: top −10 − (−4) = −6, bottom −2 − 24 = −26.
- 2a − 4b = \vec−6−26
Answer \vec−6−26
02 / Adding and multiplying column vectorsBoth tiers
a = \vec−65 and b = \vec3−1
Work out 4a + 2b as a column vector.
Show the answer and working
- Multiply both numbers in a by 4: 4a = \vec−2420
- Multiply both numbers in b by 2: 2b = \vec6−2
- Add the top numbers, then the bottom numbers: top −24 + 6 = −18, bottom 20 + (−2) = 18.
- 4a + 2b = \vec−1818
Answer \vec−1818
03 / Adding and multiplying column vectorsBoth tiers
a = \vec−15 and b = \vec4−3
Work out 4a − 3b as a column vector.
Show the answer and working
- Multiply both numbers in a by 4: 4a = \vec−420
- Multiply both numbers in b by 3: 3b = \vec12−9
- Subtract the top numbers, then the bottom numbers: top −4 − 12 = −16, bottom 20 − (−9) = 29.
- 4a − 3b = \vec−1629
Answer \vec−1629
04 / Vectors in shapes: midpointsHigher only
OAB is a triangle. OA = a and OB = 3b. Write the vector AB in terms of a and b, as simply as you can.
Show the answer and working
- To go from A to B, go back along OA, then along OB.
- AB = AO + OB = 3b − a.
- So AB = 3b − a.
Answer 3b − a
05 / Vectors in shapes: midpointsHigher only
OACB is a parallelogram. OA = a and OB = 6b. M is the midpoint of AC. Write the vector BM in terms of a and b, as simply as you can.
Show the answer and working
- AC is the same vector as OB, so AM = half of it = 3b.
- BM = BO + OA + AM = −6b + a + 3b = a − 3b.
- So BM = a − 3b.
Answer a − 3b
06 / Vectors in shapes: midpointsHigher only
OACB is a parallelogram. OA = 2a and OB = b. M is the midpoint of AC and N is the midpoint of BC. Write the vector MN in terms of a and b, as simply as you can.
Show the answer and working
- M is the midpoint of AC and N is the midpoint of BC.
- MC is half of AC: 12b. CN is half of CB: −a.
- MN = MC + CN = 12b − a.
- So MN = 12b − a.
Answer 12b − a
07 / Vectors in shapes: points in a ratioHigher only
OAB is a triangle. OA = a and OB = 2b. P is the point on AB with AP : PB = 2 : 3. Write the vector OP in terms of a and b, as simply as you can.
Show the answer and working
- AB = AO + OB = 2b − a.
- AP : PB = 2 : 3 splits AB into 5 equal parts. AP is 2 of them: AP = 25 of AB = 45b − 25a.
- OP = OA + AP = a + (45b − 25a) = 35a + 45b.
Answer 35a + 45b
08 / Vectors in shapes: points in a ratioHigher only
OAB is a triangle. OA = a and OB = b. P is the point on AB with AP : PB = 2 : 1. Write the vector OP in terms of a and b, as simply as you can.
Show the answer and working
- AB = AO + OB = b − a.
- AP : PB = 2 : 1 splits AB into 3 equal parts. AP is 2 of them: AP = 23 of AB = 23b − 23a.
- OP = OA + AP = a + (23b − 23a) = 13a + 23b.
Answer 13a + 23b
02 / What's covered3 kinds of question
Vectors at GCSE and where marks go
- Adding and multiplying column vectorsBoth tiers
- Vectors in shapes: midpointsHigher only
- Vectors in shapes: points in a ratioHigher only
Worked example 2: vectors in shapes: points in a ratio
OAB is a triangle. OA = 3a and OB = b. P is the point on AB with AP : PB = 1 : 3. Write the vector AP in terms of a and b, as simply as you can.
- AB = AO + OB = b − 3a.
- AP : PB = 1 : 3 splits AB into 4 equal parts. AP is 1 of them: AP = 14 of AB = 14b − 34a.
- So AP = 14b − 34a.
Answer 14b − 34a
A common mistake
Getting the direction wrong. If the vector from A to B is a, then the vector from B to A is −a, so going back along a route changes every sign.
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