Column vectors are written in the form (x, y), where x is the top (horizontal) number and y is the bottom (vertical) number. a = (5, -2) and b = (-3, 4).
(a)Write down 2a as a column vector.(1)
(b)Work out a + b as a column vector.(1)
(c)Work out a - b as a column vector.(1)
(Total for Question 1 is 3 marks)
2
On a centimetre grid, point A is at (2, 5) and point B is at (7, 1).
(a)Write the vector AB as a column vector.(1)
(b)Write the vector BA as a column vector.(1)
(Total for Question 2 is 2 marks)
3
Triangle PQR has vertex P at (-3, 2). Triangle PQR is translated by the vector (4, -5) to give triangle P'Q'R'.
(a)Write down the coordinates of P'.(1)
(b)Write down the column vector that translates P' back to P.(1)
(Total for Question 3 is 2 marks)
4
v is the column vector (9, 12). Calculate the magnitude of v, |v|.
(Total for Question 4 is 2 marks)
5
w is the column vector (5, 11). Work out |w|, giving your answer correct to 1 decimal place.
(Total for Question 5 is 2 marks)
6
Simplify fully 5(a + 2b) - 3(2a - b).
(Total for Question 6 is 3 marks)
7
p = 6a - 9b and q = 4a - 6b.
(a)Show that p and q are parallel.(2)
(b)Write down the ratio of the length of p to the length of q.(1)
(Total for Question 7 is 3 marks)
8
OABC is a parallelogram. OA = a and OC = c.
Diagram NOT accurately drawn
(a)Find CA in terms of a and c.(1)
(b)M is the midpoint of AB. Find OM in terms of a and c, giving your answer in its simplest form.(3)
(Total for Question 8 is 4 marks)
9
O is the origin. The position vectors of points A and B, relative to O, are a and b. M is the midpoint of AB.
Diagram NOT accurately drawn
(a)Find the vector AB in terms of a and b.(1)
(b)Find the position vector of M in terms of a and b, giving your answer in its simplest form.(2)
(Total for Question 9 is 3 marks)
10
O is the origin. The position vectors of A and B, relative to O, are a and b. P lies on AB such that AP : PB = 2 : 3. Find the position vector of P, OP, in terms of a and b, giving your answer in its simplest form.
Diagram NOT accurately drawn
(Total for Question 10 is 3 marks)
11
In triangle OAB, OA = a and OB = b. C is the point on OA such that OC = (1/3)a. D is the point on OB such that OD = (1/3)b.
Diagram NOT accurately drawn
(a)Find the vector CD in terms of a and b.(2)
(b)Hence show that CD is parallel to AB, and give the ratio CD : AB.(2)
(Total for Question 11 is 4 marks)
12
A ship sails from a harbour with a displacement vector (12, 5) km. It then changes course and sails with a displacement vector (-4, 9) km.
(a)Find the total displacement of the ship from the harbour as a column vector.(1)
(b)Calculate the total distance of the ship from the harbour, giving your answer correct to the nearest 0.1 km.(2)
(Total for Question 12 is 3 marks)
13
a + b = (7, 2). a = (3, -5). Find b as a column vector.
(Total for Question 13 is 2 marks)
14
O is the origin. OABC has position vectors OA = a, OB = a + 2b and OC = 2b. Show that OABC is a parallelogram.
Diagram NOT accurately drawn
(Total for Question 14 is 3 marks)
15
Which of the following column vectors has a magnitude of exactly 10?
A) (6, 8)
B) (5, 5)
C) (7, 7)
D) (3, 9)
(Total for Question 15 is 1 mark)
16
u = (3, -1), v = (-5, 4) and w = (2, 6).
(a)Work out u + v + w as a column vector.(1)
(b)Calculate the magnitude of u + v + w.(2)
(Total for Question 16 is 3 marks)
17
OABC is a parallelogram. OA = a and OC = c. Prove, using vectors, that the diagonals OB and AC bisect each other.
Diagram NOT accurately drawn
(Total for Question 17 is 4 marks)
18
In triangle OAB, OA = a and OB = b. E is the point on OA extended beyond A such that OE = (3/2)a. F is the point on OB extended beyond B such that OF = (3/2)b.
Diagram NOT accurately drawn
(a)Find EF in terms of a and b.(2)
(b)Show that EF is parallel to AB, and find the ratio EF : AB.(3)
(Total for Question 18 is 5 marks)
19
O is the origin. The position vectors of A and B are a and b. Q lies on the line AB extended beyond B such that AQ = (5/3)AB. Find the position vector of Q, OQ, in terms of a and b, giving your answer in its simplest form.
Diagram NOT accurately drawn
(Total for Question 19 is 3 marks)
20
OABC is a parallelogram. OA = a and OC = c. N is the point on CB such that CN : NB = 1 : 2. Lines ON and AC intersect at X.
Diagram NOT accurately drawn
(a)Find the position vector of N in terms of a and c.(2)
(b)Show that AX : XC = 3 : 1.(4)
(Total for Question 20 is 6 marks)
Mark scheme · 5.21 Vectors
Question 1
(a) B1 (10, -4) cao
(a) Answer: (10, -4)
(b) B1 (2, 2) cao
(b) Answer: (2, 2)
(c) B1 (8, -6) cao
(c) Answer: (8, -6)
Question 2
(a) B1 (5, -4) cao
(a) Answer: (5, -4)
(b) B1 (-5, 4) ft from (a)
(b) Answer: (-5, 4)
Question 3
(a) B1 (1, -3) cao
(a) Answer: (1, -3)
(b) B1 (-4, 5) ft from (a)
(b) Answer: (-4, 5)
Question 4
M1√92 + 122 oe
A1 15 cao
Answer: 15
Question 5
M1√52 + 112 oe
A1 awrt 12.1
Answer: 12.1
Question 6
M1 expands both brackets: 5a + 10b - 6a + 3b oe
M1 collects a terms and b terms correctly (dependent on at least one bracket expanded)
A1 -a + 13b oe, e.g. 13b - a
Answer: -a + 13b
Question 7
(a) M1 writes p or q as a scalar multiple of the other, e.g. p = 1.5 x q oe
(a) A1 cso: states p = (3/2)q (or q = (2/3)p), so p and q are parallel
(a) Answer: p = (3/2)q so p and q are parallel
(b) B1 3 : 2 oe (ft from the scalar in (a))
(b) Answer: 3 : 2
Question 8
(a) B1 a - c oe
(a) Answer: CA = a - c
(b) M1 identifies OB = a + c (since AB = OC = c in a parallelogram)
(b) M1 OM = OA + (1/2)AB, or OM = (1/2)(OA + OB), correct method
(b) A1 a + (1/2)c oe, e.g. (2a + c)/2
(b) Answer: OM = a + (1/2)c
Question 9
(a) B1 b - a cao
(a) Answer: AB = b - a
(b) M1 OM = a + (1/2)(b - a) or OM = (1/2)(a + b), correct method
(b) A1 (1/2)(a + b) oe, e.g. (a+b)/2
(b) Answer: OM = (a + b)/2
Question 10
M1 AB = b - a
M1 OP = a + (2/5)(b - a), correct method for the ratio
A1 (3a + 2b)/5 oe, e.g. (3/5)a + (2/5)b
Answer: OP = (3a + 2b)/5
Question 11
(a) M1 CD = OD - OC = (1/3)b - (1/3)a
(a) A1 (1/3)(b - a) oe, e.g. (b-a)/3
(a) Answer: CD = (1/3)(b - a)
(b) M1 states AB = b - a
(b) A1 cso: CD = (1/3)AB so CD is parallel to AB, ratio CD : AB = 1 : 3
(b) Answer: CD is parallel to AB; CD : AB = 1 : 3
Question 12
(a) B1 (8, 14) cao
(a) Answer: (8, 14)
(b) M1√82 + 142 ft from (a)
(b) A1 awrt 16.1
(b) Answer: 16.1 km
Question 13
M1 b = (7, 2) - (3, -5)
A1 (4, 7) cao
Answer: (4, 7)
Question 14
M1 finds AB = OB - OA = 2b
M1 compares AB with OC = 2b
A1 cso: AB = OC, so AB is parallel and equal to OC, so OABC is a parallelogram
Answer: AB = OC = 2b, so OABC is a parallelogram
Question 15
B1 A cao
Answer: A) (6, 8)
Question 16
(a) B1 (0, 9) cao
(a) Answer: (0, 9)
(b) M1√02 + 92 ft from (a)
(b) A1 9 cao
(b) Answer: 9
Question 17
M1 OB = a + c, so the midpoint of OB has position vector (1/2)(a + c)
M1 midpoint of AC has position vector a + (1/2)(c - a)
A1 simplifies midpoint of AC to (1/2)(a + c)
A1 cso: both diagonals have the same midpoint, (1/2)(a+c), so the diagonals bisect each other
Answer: Midpoint of OB = midpoint of AC = (1/2)(a + c), so the diagonals bisect each other
Question 18
(a) M1 EF = OF - OE = (3/2)b - (3/2)a
(a) A1 (3/2)(b - a) oe
(a) Answer: EF = (3/2)(b - a)
(b) M1 states AB = b - a
(b) M1 EF = (3/2)AB (ft from part a)
(b) A1 cso: EF is a scalar multiple of AB so they are parallel; ratio EF : AB = 3 : 2
(b) Answer: EF is parallel to AB; EF : AB = 3 : 2
Question 19
M1 AB = b - a
M1 OQ = a + (5/3)(b - a), correct method
A1 (5b - 2a)/3 oe, e.g. -(2/3)a + (5/3)b
Answer: OQ = (5b - 2a)/3
Question 20
(a) M1 identifies CB = a (since CB = OA = a in the parallelogram) and N = OC + (1/3)CB
(a) A1 c + (1/3)a oe
(a) Answer: ON = c + (1/3)a
(b) M1 writes a point on ON as t(c + (1/3)a) for parameter t
(b) M1 writes a point on AC as a + u(c - a) for parameter u
(b) M1 equates coefficients of a and c to form and solve simultaneous equations, e.g. t/3 = 1-u and t = u
(b) A1 cso: solves to t = u = 3/4, so X divides AC with AX:XC = 3:1