Given a = (-11) and b = (-45)
Work out a + b
Work out a + b
Worked Solution & Answer
(-56) B1
Answer: (-56)
Vectors — Notes & Worked Examples Show / Hide
Explanation:
A column vector writes horizontal movement on top (+ right, − left) and vertical movement on the bottom (+ up, − down). In geometric vector diagrams, going against an arrow reverses the sign (BA = −AB), and to travel from A to B via O you add the vectors: AB = AO + OB = −a + b = b − a.
Two vectors are parallel if one is a scalar multiple of the other (for example, 2a + 4b = 2(a + 2b) is parallel to a + 2b).
Example 1: Given column vectors a = (4, −1) and b = (−2, 5), work out 3a − 2b.
Solution:
Multiply each component: 3a = (12, −3) and 2b = (−4, 10)
Top component: 12 − (−4) = 16
Bottom component: −3 − 10 = −13
Answer: (16, −13) as a column vector
Example 2: In ▵OAB, OA = 2a and OB = 4b. M is the midpoint of AB. Find OM in terms of a and b.
Solution:
AB = AO + OB = −2a + 4b
AM = 12AB = −a + 2b
OM = 2a + (−a + 2b)
Answer: a + 2b
Given c = (4-2)
Work out 2c
Work out 2c
Worked Solution & Answer
(8-4) B1
Answer: (8-4)
Vector a = (32) and vector b = (1-4).
(a) On the grid, draw vector a. [1 mark]
(b) On the grid, draw vector a + b. [2 marks]
Worked Solution & Answer
(a)
Vector a goes 3 right and 2 up. B1
Answer (a): Directed line segment going 3 units right and 2 units up with direction arrow.
(b)
a + b = (3 + 1, 2 − 4) = (4, −2): 4 right and 2 down. M1 A1
Answer (b): Directed line segment going 4 units right and 2 units down with direction arrow.
Vector **a** is drawn on this grid.
(a) Write vector **a** as a column vector. [2 marks]
(b) Write this vector in terms of **a**. [1 mark]
Worked Solution & Answer
(a)
Vector a goes 2 units right and 3 units down M1
⇒ (2, −3). A1
⇒ (2, −3). A1
Answer (a): (2-3)
(b)
Same length and parallel to a, but in the opposite direction ⇒ −a. B1
Answer (b): -a
The vector **a** and the vector **b** are shown on the grid.
(a) On the grid, draw and label vector -2a [1 mark]
(b) Work out a + 2b as a column vector. [2 marks]
Worked Solution & Answer
(a)
a = (1, 2) ⇒ –2a = (–2, –4) B1
Answer (a): Vector drawn going 2 units left and 4 units down with arrow
(b)
a + 2b = (1, 2) + 2(1, –3) = (1 + 2, 2 – 6) = (3, –4) M1 A1
Answer (b): 3, –4
p = (54) and q = (1-1.5)
(a) Work out the column vector 2p - q. [2 marks]
(b) The grid shows the vector q.
On the same grid, draw the vector -3q. [2 marks]
On the same grid, draw the vector -3q. [2 marks]
Worked Solution & Answer
(a)
2p - q = 2(54) - (1-1.5) = (108) - (1-1.5) = (99.5). M1 A1
Answer (a): (99.5)
(b)
-3q = -3(1-1.5) = (-34.5) M1
From start point, move 6 grid squares left and 9 grid squares up, with arrow pointing top-left. A1
From start point, move 6 grid squares left and 9 grid squares up, with arrow pointing top-left. A1
Answer (b): Vector of length 3 times q pointing in the opposite direction (-34.5).
Vectors **a** and 4b are drawn on the grid.
(a) Write vector **a** as a column vector. [2 marks]
(b) Find vector **b** as a column vector. [2 marks]
Worked Solution & Answer
(a)
4 squares right and 2 squares down M1
⇒ (4, −2). A1
⇒ (4, −2). A1
Answer (a): (4-2)
(b)
4b = (4, 9), so b = (4 ÷ 4, 9 ÷ 4) = (1, 9/4) or (1, 2.25). M1 A1
Answer (b): (12.25)
a = (23) b = (-12) c = (41)
(a) a + b [1 mark]
(b) 2a - c [2 marks]
(b) From the point P, draw the vector 2d [1 mark]
Worked Solution & Answer
(a)
(2 + –1, 3 + 2) = (1, 5) B1
Answer (a): (15)
(b)
(4 – 4, 6 – 1) = (0, 5) M1 A1
Answer (b): (05)
(b)
d = (3, –2) ⇒ 2d = (6, –4) B1
Answer (b): Vector drawn from P going 6 squares right and 4 squares down
The diagram shows the vectors **a** and **b**.
As a column vector a = (32)
As a column vector a = (32)
(a) What is **b** as a column vector? [2 marks]
Answer
(b) Work out 4a as a column vector. [1 mark]
Answer
(c) a + c = (30)
Work out **c** as a column vector.
Circle your answer.
(20) (02) (-20) (0-2) [1 mark]
Work out **c** as a column vector.
Circle your answer.
(20) (02) (-20) (0-2) [1 mark]
Worked Solution & Answer
(a)
b goes 4 right and 1 down = (4, –1) M1 A1
Answer (a): Answer (4-1)
(b)
4 × (3, 2) = (12, 8) B1
Answer (b): Answer (128)
(c)
c = (3, 0) – (3, 2) = (0, –2) B1
Answer (c): (0-2)
Vector a and vector 2b are drawn on this grid.
(a) Write vector a as a column vector. [2 marks]
(b) Write a column vector that is different in direction but has the same length as vector a. [1 mark]
(c) On the grid below, draw the vector a - b. [3 marks]
Worked Solution & Answer
(a)
Vector a goes 3 units left and 1 unit up: (-3, 1). M1 A1
Answer (a): (-31)
(b)
Any column vector (p, q) ≠ (-3, 1) with p² + q² = 10, e.g
(3, 1), (1, 3), or (3, -1). B1
(3, 1), (1, 3), or (3, -1). B1
Answer (b): (31)
(c)
2b = (0, -4) ⇒ b = (0, -2). M2
a − b = (-3, 1) − (0, -2) = (-3, 3). A1
a − b = (-3, 1) − (0, -2) = (-3, 3). A1
Answer (c): Vector (-3, 3) drawn on grid with arrow pointing up-left