Edexcel GCSE Maths · G24–G25

Vectors

50 maths questions with worked answers with answers for G24–G25 Vectors in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

All 50 questions and answers

Question 1 · Column vectors · G24
What does the column vector 3−2 represent?
Answer:
A movement.
3 units right and 2 units down.
Question 2 · Column vectors · G24
Write the column vector for a movement of 4 left and 5 up.
Answer:
Left is negative x; up is positive y.
−45.
Question 3 · Adding vectors · G25
a = 25 and b = 3−1. Find a + b.
Answer:
Add the top and bottom numbers separately.
54.
Question 4 · Subtracting vectors · G25
a = 61 and b = 24. Find a − b.
Answer:
6 − 2 = 4 and 1 − 4 = −3.
4−3.
Question 5 · Scalar multiples · G25
a = 3−2. Find 4a.
Answer:
Multiply each component by 4.
12−8.
Question 6 · Combining · G25
a = 13 and b = −20. Find 2a + 3b.
Answer:
2a = 26 and 3b = −60.
−46.
Question 7 · Combining · G25
p = 52 and q = 1−3. Find 3p − 2q.
Answer:
3p = 156 and 2q = 2−6.
1312.
Question 8 · Vectors and points · G24
A is (2, 3) and B is (7, 1). Write the vector AB.
Answer:
Subtract the coordinates of A from B: 7 − 2 = 5 and 1 − 3 = −2.
AB = 5−2.
Question 9 · Vectors and points · G24
Point P is (1, −4). Translate it by −36.
Answer:
1 − 3 = −2 and −4 + 6 = 2.
(−2, 2).
Question 10 · Diagrams · G24
How is a vector shown on a diagram?
Answer:
A line segment with an arrow.
The arrow shows the direction; the length shows the magnitude.
Question 11 · Notation · G24
How is a vector written in print and by hand?
Answer:
In print it is bold: a.
By hand it is underlined: a̲. A vector from A to B is written AB with an arrow above.
Question 12 · Negative vectors · G24
If AB = a, what is BA?
Answer:
BA goes in the opposite direction.
BA = −a.
Question 13 · Magnitude · G25 · Higher only
Find the magnitude of 68.
Answer:
Use Pythagoras: √(6² + 8²) = √100.
10.
Question 14 · Magnitude · G25 · Higher only
Find the magnitude of −512.
Answer:
√(25 + 144) = √169.
13.
Question 15 · Parallel vectors · G25
Are 23 and 69 parallel? Explain.
Answer:
69 = 3 × 23.
Yes; one is a multiple of the other.
Question 16 · Parallel vectors · G25
Are 4−2 and −21 parallel?
Answer:
4−2 = −2 × −21.
Yes; they are parallel but point in opposite directions.
Question 17 · Parallel vectors · G25 · Higher only
a = 3k is parallel to 610. Find k.
Answer:
610 = 2 × 35.
k = 5.
Question 18 · Vector paths · G25
OA = a and OB = b. Find AB.
Answer:
AB = AO + OB.
AB = −a + b = b − a.
Question 19 · Vector paths · G25
OA = a and OB = b. M is the midpoint of AB. Find AM.
Answer:
AB = b − a.
AM = 12(b − a).
Question 20 · Vector paths · G25 · Higher only
OA = a and OB = b. M is the midpoint of AB. Find OM.
Answer:
OM = OA + AM = a + 12(b − a).
OM = 12a + 12b = 12(a + b).
Question 21 · Ratio on a line · G25 · Higher only
OA = a and OB = b. P lies on AB with AP : PB = 1 : 2. Find OP.
Answer:
AP = 13AB = 13(b − a).
OP = a + 13(b − a) = 23a + 13b.
Question 22 · Ratio on a line · G25 · Higher only
OA = a and OB = b. P lies on AB with AP : PB = 3 : 1. Find OP.
Answer:
AP = 34(b − a).
OP = a + 34(b − a) = 14a + 34b.
Question 23 · Collinear points · G25 · Higher only
How do you prove three points A, B and C are collinear using vectors?
Answer:
Show that AB is a multiple of BC (or AC).
They share point B, so the points lie on a straight line.
Question 24 · Collinear points · G25 · Higher only
AB = 2a + 4b and BC = a + 2b. Show that A, B and C are collinear.
Answer:
AB = 2(a + 2b) = 2BC.
AB is parallel to BC and they share B, so A, B and C are collinear.
Question 25 · Parallelograms · G25 · Higher only
In parallelogram OABC, OA = a and OC = c. Find OB.
Answer:
OB = OA + AB, and AB = OC = c.
OB = a + c.
Question 26 · Parallelograms · G25 · Higher only
In parallelogram OABC, OA = a and OC = c. Find AC.
Answer:
AC = AO + OC.
AC = c − a.
Question 27 · Proof · G25 · Higher only
Show that the line joining the midpoints of two sides of a triangle is parallel to the third side.
Answer:
In triangle OAB, OA = a and OB = b. Midpoints: OM = 12a and ON = 12b.
MN = 12b − 12a = 12(b − a) = 12AB, so MN is parallel to AB and half its length.
Question 28 · Hexagons · G25 · Higher only
In a regular hexagon ABCDEF with centre O, OA = a and OB = b. Find AB.
Answer:
AB = AO + OB.
AB = b − a.
Question 29 · Hexagons · G25 · Higher only
In regular hexagon ABCDEF with centre O, OA = a and OB = b. Find OC.
Answer:
OC is parallel to AB and the same length (the triangles are equilateral).
OC = b − a.
Question 30 · Hexagons · G25 · Higher only
In regular hexagon ABCDEF with centre O, OA = a and OB = b. Find AD.
Answer:
AD passes through O and is twice AO.
AD = −2a.
Question 31 · Equal vectors · G25
When are two vectors equal?
Answer:
When they have the same magnitude and the same direction.
Their position does not matter.
Question 32 · Unknown scalars · G25 · Higher only
OP = 2a + b and OQ = ka + 3b. P, Q and O are collinear. Find k.
Answer:
OQ must be a multiple of OP: 3b = 3 × b, so OQ = 3OP.
k = 3 × 2 = 6.
Question 33 · Resultant · G25
A boat moves 41 then 25. Find its resultant movement.
Answer:
Add the vectors.
66.
Question 34 · Resultant · G25 · Higher only
Find the magnitude of the resultant 66, to 1 d.p.
Answer:
√(36 + 36) = √72.
8.5.
Question 35 · Solving with vectors · G25
x3 + 4y = 9−1. Find x and y.
Answer:
x + 4 = 9 and 3 + y = −1.
x = 5, y = −4.
Question 36 · Solving with vectors · G25
2a5 = 8b. Find a and b.
Answer:
2a = 8 and 2 × 5 = b.
a = 4, b = 10.
Question 37 · Vector paths · G25 · Higher only
In triangle OAB, OA = a and OB = b. X lies on OA with OX = 23OA. Find XB.
Answer:
OX = 23a.
XB = XO + OB = −23a + b.
Question 38 · Vector paths · G25 · Higher only
OA = 3a, OB = 3b. P divides OA in the ratio 1 : 2. Find PB.
Answer:
OP = a.
PB = −a + 3b = 3b − a.
Question 39 · Proof · G25 · Higher only
OA = a, OB = b. C is such that OC = 2b − a. Show that B is the midpoint of AC.
Answer:
AB = b − a and BC = OC − OB = 2b − a − b = b − a.
AB = BC, so B is the midpoint of AC.
Question 40 · Parallel lines · G25 · Higher only
AB = 3a − 6b and CD = −a + 2b. What can you say about AB and CD?
Answer:
AB = −3(−a + 2b) = −3CD.
AB is parallel to CD and 3 times as long.
Question 41 · Translations · G24
What vector translates (6, 2) to (1, 5)?
Answer:
1 − 6 = −5 and 5 − 2 = 3.
−53.
Question 42 · Translations · G24
A shape is translated by 30. Describe the movement.
Answer:
The bottom number is 0.
3 units to the right, with no vertical movement.
Question 43 · Subtracting vectors · G25
a = −14. Find −3a.
Answer:
Multiply each component by −3.
3−12.
Question 44 · Magnitude · G25 · Higher only
Find the magnitude of 23−4.
Answer:
23−4 = 6−8.
√(36 + 64) = 10.
Question 45 · Unknown scalars · G25 · Higher only
ma + nb = 4a − 7b, where a and b are not parallel. Find m and n.
Answer:
Compare coefficients of a and b.
m = 4, n = −7.
Question 46 · Combining · G25
Simplify 3(a + 2b) − 2(a − b).
Answer:
3a + 6b − 2a + 2b.
a + 8b.
Question 47 · Vector paths · G25 · Higher only
OABC is a trapezium with OA = a, OC = c and CB = 2a. Find AB.
Answer:
AB = AO + OC + CB = −a + c + 2a.
AB = a + c.
Question 48 · Zero vector · G24
What is a + (−a)?
Answer:
The movements cancel out.
The zero vector 00.
Question 49 · Scalar multiples · G25
What does multiplying a vector by a negative number do?
Answer:
It reverses the direction.
It also scales the length by the size of the number.
Question 50 · Proof · G25 · Higher only
Points A, B and C have AB = 4p and BC = 6p. Find the ratio AB : BC and explain what it shows.
Answer:
AB : BC = 4 : 6 = 2 : 3.
They are parallel and share B, so A, B and C are collinear, with B dividing AC in the ratio 2 : 3.

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