A-Level Maths (Edexcel 9MA0) · Mechanics Year 1 · Chapter 8: Modelling in mechanics · Lesson 2

Working with vectors

125 marks

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Date

Do Now

Quick questions to start. Some use skills from GCSE and Pure Year 1; some set slips that cost marks.

A1
Needed today
Find the magnitude of $7 i - 24 j$ and a unit vector in its direction.
[2 marks]
Answer
A2
Needed today
A point $B$ is $8$ km from $A$ on a bearing of $15 0^{\circ}$ . Find how far east and how far south of $A$ the point $B$ is.
[3 marks]
Answer
A3
Common slip
Write $45 km h^{- 1}$ in $m s^{- 1}$ .
[2 marks]
Answer
A4
Common slip
A particle moves along a straight line. The positive direction is to the right. Its velocity is $- 6 m s^{- 1}$ . Ben says that its speed is $- 6 m s^{- 1}$ . Correct him, and state the direction in which the particle moves.
[2 marks]
Answer
A5
Common slip
Kai says that the angle between $- 3 i + 4 j$ and $i$ is $53. 1^{\circ}$ . Find the correct angle.
[3 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

A cyclist rides along a straight road. East is the positive direction. She starts at $O$ , rides $400$ m east at $5 m s^{- 1}$ , then turns and rides at $4 m s^{- 1}$ to a point $150$ m west of $O$ .
(a) Find the time for each part of the ride.
(b) Find the total distance travelled and the final displacement from $O$ .
(c) Find the average speed and the average velocity for the whole ride.

Hint Distance and speed are scalars; displacement and velocity are vectors and keep their sign.

B1
A runner runs along a straight path. North is the positive direction. She starts at $O$ , runs $1200$ m north at $8 m s^{- 1}$ , then turns and runs back at $6 m s^{- 1}$ until she is $600$ m north of $O$ .
(a) Write down her velocity on the way back.
(b) Find the total distance she runs and her final displacement from $O$ .
(c) Find her average speed and her average velocity.
[6 marks]
Answer

Example 2

A particle moves with velocity $(- 5 i + 12 j) m s^{- 1}$ , where $i$ is east and $j$ is north. Find its speed, the angle between its direction of motion and $i$ , and the bearing on which it moves.

Hint Always sketch the vector first. The angle with i is measured from the positive i direction; a bearing is measured clockwise from north.

B2
A particle moves with velocity $(6 i - 8 j) m s^{- 1}$ , where $i$ is east and $j$ is north. Find its speed, the angle between its direction of motion and $i$ , and the bearing on which it moves.
[4 marks]
Answer

Example 3

(a) A ship sails at $15 km h^{- 1}$ on a bearing of $21 0^{\circ}$ . Write its velocity in terms of $i$ (east) and $j$ (north).
(b) A particle has velocity $(p i + 4 j) m s^{- 1}$ and speed $5 m s^{- 1}$ . Find the possible values of $p$ .
(c) A particle moves parallel to $4 i - 3 j$ with speed $10 m s^{- 1}$ . Find its velocity.

Hint A direction vector fixes the direction only: scale it to the speed.

B3
(a) A boat moves at $20 km h^{- 1}$ on a bearing of $30 0^{\circ}$ . Write its velocity in terms of $i$ (east) and $j$ (north).
(b) A particle has velocity $(3 i + q j) m s^{- 1}$ and speed $35 m s^{- 1}$ . It moves on a bearing between $09 0^{\circ}$ and $18 0^{\circ}$ . Find $q$ .
(c) A particle moves parallel to $5 i + 12 j$ with speed $26 m s^{- 1}$ . Find its velocity.
[5 marks]
Answer

Example 4

A drone flies from $A$ to $B$ , where $A B = (8 i + 6 j)$ m, and then from $B$ to $C$ , where $B C = (5 i - 18 j)$ m.
(a) Find $A C$ and the distance $A C$ .
(b) Find the total distance the drone flies.
(c) Find the angle between $A C$ and $i$ .

Hint Add the displacements to find where it ends; add the lengths to find how far it flew.

B4
A hiker walks from $A$ to $B$ , where $A B = (- 9 i + 12 j)$ km, and then from $B$ to $C$ , where $B C = (15 i + 8 j)$ km. Here $i$ is east and $j$ is north.
(a) Find $A C$ and the distance $A C$ .
(b) Find the total distance walked.
(c) Find the bearing of $C$ from $A$ .
[6 marks]
Answer

Example 5

A particle moves with constant velocity. At $t = 0$ it is at the point with position vector $(2 i - 3 j)$ m, and at $t = 3$ s it is at $(14 i + 6 j)$ m.
(a) Find its velocity and speed.
(b) Find the time when it is due east of $O$ , and its position then.

Hint r = r₀ + vt. "Due east of O" means the j-part is zero AND the i-part is positive.

B5
A ship moves with constant velocity. At noon it is at $(- 8 i + 5 j)$ km and at $3$ pm it is at $(4 i - 4 j)$ km, where $i$ is east and $j$ is north, relative to a port $O$ .
(a) Find its velocity and speed.
(b) Find the time when it is due south of $O$ , and its distance from $O$ then.
(c) Show that the ship is never due east of $O$ .
[6 marks]
Answer

Practice

A-Level

C1
A boat sails $(24 i + 7 j)$ km and then $(- 12 i + 9 j)$ km. Sam says: "The boat has travelled $25 + 15 = 40$ km, so it is now $40$ km from where it started." Explain Sam's mistake and find the distance of the boat from its start.
[4 marks]
Answer

C2
A particle moves with velocity $(- 4 i - 3 j) m s^{- 1}$ , where $i$ is east and $j$ is north. Ana says: " $tan^{- 1} \frac{3}{4} = 36. 9^{\circ}$ , so the angle with $i$ is $36. 9^{\circ}$ and the bearing is $036. 9^{\circ}$ ." Find the correct angle with $i$ and the correct bearing.
[4 marks]
Answer
C3
A particle starts at the point with position vector $(2 i + j)$ m. It moves with constant velocity in the direction of $3 i - 4 j$ with speed $15 m s^{- 1}$ . Jo says that its velocity is $(3 i - 4 j) m s^{- 1}$ . Explain why Jo is wrong, and find the position vector of the particle after $4$ s.
[5 marks]
Answer
C4
A particle moves along a straight line with constant acceleration. Its velocity changes from $6 m s^{- 1}$ to $- 4 m s^{- 1}$ . Kai says: "Its speed fell by $10 m s^{- 1}$ ." Explain why Kai is wrong.
[4 marks]
Answer
C5
A boat moves at $12 km h^{- 1}$ on a bearing of $12 0^{\circ}$ , with $i$ east and $j$ north. Lee writes the velocity as $(12 cos 12 0^{\circ}) i + (12 sin 12 0^{\circ}) j = (- 6 i + 10.4 j) km h^{- 1}$ . Find the correct velocity, and the bearing on which Lee's velocity points.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
In this question $i$ is east, $j$ is north and distances are in km. Two ships $P$ and $Q$ move with constant velocities. At noon $P$ is at $(2 i + 10 j)$ and $Q$ is at $(- 6 i - 2 j)$ . The velocity of $P$ is $(4 i - 2 j) km h^{- 1}$ and the velocity of $Q$ is $(6 i + 2 j) km h^{- 1}$ . At time $t$ hours after noon the position vectors of $P$ and $Q$ are $p$ and $q$ .
(a)
Find $p$ and $q$ in terms of $t$ .
[2 marks]
Answer
(b)
Show that the ships do not collide.
[3 marks]
Answer
(c)
Find the distance between the ships at $2$ pm.
[3 marks]
Answer
(d)
Find the time when $Q$ is due north of $P$ , and the distance between them then.
[3 marks]
Answer
Total for D1: 11 marks

D2
A ship moves with constant velocity $(- 8 i + 15 j) km h^{- 1}$ , where $i$ is east and $j$ is north, relative to a fixed point $O$ . Mo says: " $tan^{- 1} \frac{15}{8} = 61. 9^{\circ}$ , so the ship sails on a bearing of $061. 9^{\circ}$ ."
Mo is wrong.
(a)
Find the speed of the ship.
[2 marks]
Answer
(b)
Explain Mo's mistake and find the bearing of the ship's course.
[3 marks]
Answer
(c)
At $9$ am the ship is at the point $(4 i + 2 j)$ km. Find the time when it has sailed $51$ km.
[2 marks]
Answer
(d)
Find the distance of the ship from $O$ at that time.
[2 marks]
Answer
Total for D2: 9 marks
D3
A ball rolls across a smooth horizontal floor with constant velocity. At time $t = 0$ it is at the point with position vector $(- 3 i + 4 j)$ m relative to a fixed point $O$ , and at $t = 2$ s it is at $(5 i - 2 j)$ m. At $t = 0$ the ball is $5$ m from $O$ . Find the other time when the ball is $5$ m from $O$ .
[7 marks]
Answer
D4
A walker walks $3$ km on a bearing of $04 0^{\circ}$ and then $5$ km on a bearing of $13 0^{\circ}$ . Take $i$ east and $j$ north.
(a)
Write each displacement in terms of $i$ and $j$ .
[4 marks]
Answer
(b)
Find the distance and the bearing of the walker's final position from the start.
[4 marks]
Answer
(c)
Explain how the distance in (b) can be found without using components.
[2 marks]
Total for D4: 10 marks
D5
A particle $P$ moves with constant velocity. At $t = 0$ it is at $A$ , with position vector $(i - 2 j)$ m, and at $t = 4$ s it is at $B$ , with position vector $(13 i + 14 j)$ m.
(a)
Find the velocity and the speed of $P$ .
[3 marks]
Answer
(b)
Later $P$ passes through the point $D$ with position vector $(k i + 26 j)$ m. Find the time when this happens and the value of $k$ .
[3 marks]
Answer
(c)
A second particle $Q$ moves with constant velocity from the point $(25 i - 2 j)$ m at $t = 0$ , and reaches $D$ at the same time as $P$ . Find the velocity and the speed of $Q$ .
[4 marks]
Answer
Total for D5: 10 marks
D6
A model boat starts at a fixed point $O$ . It moves with velocity $(8 i + 6 j) m s^{- 1}$ for $20$ s, then with velocity $(- 5 i + 12 j) m s^{- 1}$ for $30$ s.
(a)
Find the position vector of the boat at the end of its journey.
[3 marks]
Answer
(b)
Find the total distance travelled by the boat, and its final distance from $O$ .
[3 marks]
Answer
(c)
Find the magnitude of the average velocity of the boat for the whole journey.
[2 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A drone starts at a fixed point $O$ . It flies with velocity $(5 i + 2 j) m s^{- 1}$ for $T_{1}$ s and then with velocity $(- 3 i + 4 j) m s^{- 1}$ for $T_{2}$ s. It ends due north of $O$ , $260$ m from $O$ . Its battery allows at most $400$ m of flight. Find $T_{1}$ and $T_{2}$ , decide whether the drone can make this flight, and find its average velocity.
[10 marks]
Hint 1 · What to try
Write the final position as $T_{1} (5 i + 2 j) + T_{2} (- 3 i + 4 j)$ . Due north means the $i$ part is zero.
Hint 2 · The first line
The $j$ part is $260$ . Solve the two equations for $T_{1}$ and $T_{2}$ .
Hint 3 · The full method
The distance flown is speed $\times$ time on each part, not $260$ m. The average velocity uses the displacement and the total time $T_{1} + T_{2}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
Tell a vector from a scalar, and give a speed (never negative) from a velocity.
Turn a speed and a bearing into i, j form, and back again.
Find a resultant displacement, a distance travelled and an average velocity.
Use r = r₀ + vt for an object moving with constant velocity.

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