Question 1 Easy
Vector p = (1, 2) and q = (4, 8). Find k with q = k p.
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Answer: 4
- 4 divided by 1, and check the second component.
- k = 4.
- Method (1): 4 divided by 1, and check the second component.
- Accuracy (1): Answer 4
Vector proof questions practice questions with answers
A second set of vector proof questions questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.
More questions
Question 2 Easy
A journey goes along a and then along b. Write the single vector for the whole journey.
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Answer: a+b
- Add the two steps one after the other.
- a + b.
- Method (1): Add the two steps one after the other.
- Accuracy (1): Answer a+b
Question 3 Medium
OC = 3a - b and OD = a + b. Write the vector CD.
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Answer: -2a+2b
- CD = OD - OC.
- (a + b) - (3a - b) = -2a + 2b.
- Method (1): CD = OD - OC.
- Accuracy (1): Answer -2a+2b
Question 4 Medium
OA = a, OB = b, and N divides AB in the ratio 1:1. That is the midpoint. Write ON.
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Answer: 1/2(a+b)
- Ratio 1:1 is the midpoint.
- 1/2(a+b).
- Method (1): Ratio 1:1 is the midpoint.
- Accuracy (1): Answer 1/2(a+b)
Question 5 Hard
P = (3, 6) and Q = (9, 18). Give the scalar k with Q = k P, which also shows that O, P and Q are collinear.
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Answer: 3
- 9 / 3 = 3.
- The shared point is the origin, so a positive scale puts Q on the ray OP.
- Method (1): 9 / 3 = 3.
- Accuracy (1): Answer 3
Question 6 Hard
Write the vector from the point with position a + b to the point with position 3a + b.
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Answer: 2a
- Subtract the start.
- 3a + b - a - b = 2a.
- Method (1): Subtract the start.
- Accuracy (1): Answer 2a
Exam-style question
A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.
Exam-style question 14 marks
OA = a and OB = b. M is the midpoint of AB. Write OM, and write AB.
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Answer: 1/2(a+b) and b-a
- OM = 1/2(a+b).
- AB = b - a.
- Both results come from the position vectors.
- The half is only on the midpoint.
- Method (1): Midpoint method
- Accuracy (1): OM
- Method (1): AB = OB - OA
- Accuracy (1): b-a
Quick checks
Short one-step questions. Pick an option before you open the answer.
Check 1
How do you write the midpoint of A and B when OA = a and OB = b?
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1/2(a+b). 1/2(a+b).
Check 2
What is vector AB if OA = a and OB = b?
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b - a. Subtract the start from the end. b - a. Subtract the start from the end.
Check 3
When are two vectors parallel?
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When one is a scalar multiple of the other. When one is a scalar multiple of the other.
Check 4
A negative scalar means what?
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The vectors are parallel but point in opposite senses. The vectors are parallel but point in opposite senses.
Check 5
Why does a shared starting point plus a scalar multiple give collinear points?
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The second point lies on the line through the first point in that vector direction. The second point lies on the line through the first point in that vector direction.
Check 6
What is wrong with answering 4 when the question asked for a position vector?
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A position needs both components, or an expression in a and b. A position needs both components, or an expression in a and b.
Check 7
Simplify the journey from a to 3a.
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2a. You subtract the start. 2a. You subtract the start.
Check 8
OC = 3a - b. What are the coefficients of a and of b?
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3 and -1. 3 and -1.
Check 9
If both components of AC are 3 times both components of AB, what is k?
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3. AC = 3 AB. 3. AC = 3 AB.
Check 10
Why halve a + b for a midpoint?
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The midpoint is the average of the two position vectors. The midpoint is the average of the two position vectors.
Check 11
Vector 2a + 2b is parallel to which simpler vector?
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a + b. The scalar is 2. a + b. The scalar is 2.
Check 12
Does k = 1 mean the same vector?
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Yes. The arrows are equal. Yes. The arrows are equal.
Check 13
From C to D, which vector do you subtract?
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Subtract the position of C from the position of D. Subtract the position of C from the position of D.
Check 14
OA and OB are not drawn from each other. Where do both arrows start?
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At the origin, unless the question says otherwise. At the origin, unless the question says otherwise.
Check 15
Name one conclusion you may write after you show CD = -2a.
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CD is parallel to a. CD is parallel to a.
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