A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 11: Vectors · Lesson 3

Geometric problems and modelling with vectors

70 marks

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Date

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Quick questions to start.

A1
Needed today
$A$ is $(2, - 1)$ and $B$ is $(14, 8)$ . Find the coordinates of the point $P$ on $A B$ with $A P : P B = 1 : 2$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A2
Last lesson
Find a unit vector in the direction of $20 i - 48 j$ .
[2 marks]
Answer
A3
GCSE Higher
A ship sails $6$ km due east and then $8$ km due south. Find the bearing of its finishing point from its start, to the nearest degree.
[2 marks]
Answer $\dots \dots^{\circ}$
A4
From chapter 3
Solve $2 λ = 1 - μ$ and $λ = 3 μ$ .
[2 marks]
Answer $λ = \dots\dots$ $μ = \dots\dots$

Example, then your turn

A-Level

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

Example 1

$O A = a$ , $O B = b$ , $P$ is on $A B$ with $A P : P B = 3 : 2$ . (a) Find $O P$ . (b) $O X = 2 a + 3 b$ . Show that $O$ , $P$ , $X$ are collinear.

Hint

O P = O A + \frac{3}{5} A B

. Collinear: parallel, with a point in common.

B1
$O A = a$ , $O B = b$ , $P$ is on $A B$ with $A P : P B = 1 : 3$ . (a) Find $O P$ . (b) $O Y = 6 a + 2 b$ . Show that $O$ , $P$ , $Y$ are collinear.
[4 marks]
Answer(a) $O P = \dots\dots$

Example 2

$O A = a$ , $O B = b$ , $M$ is the midpoint of $O A$ and $O N : N B = 2 : 1$ . $A N$ and $B M$ meet at $X$ . Find $O X$ and the ratio $A X : X N$ .

Hint Reach

X

along

A N

and along

B M

, each with its own unknown. Compare coefficients of

a

and

b

B2
In triangle $O A B$ , $O A = a$ , $O B = b$ , $O P : P A = 1 : 2$ and $Q$ is the midpoint of $O B$ . $A Q$ and $B P$ meet at $X$ . Find $O X$ .
[5 marks]
Answer $O X = \dots\dots$

Example 3

A walker goes $8$ km on a bearing of $06 0^{\circ}$ to $A$ , then $5$ km on $15 0^{\circ}$ to $B$ . With $i$ east, $j$ north, find $O B$ , $O B$ and the bearing of $B$ from $O$ .

Hint A leg

d

on bearing

θ

d sin θ i + d cos θ j

B3
A ship sails $12$ km on a bearing of $04 0^{\circ}$ to $A$ , then $7$ km on $13 0^{\circ}$ to $B$ . Find $O B$ ( $3$ s.f.), $O B$ , and the bearing of $B$ from $O$ ( $1$ d.p.).
[5 marks]
Answer $O B = \dots\dots$ km $\dots \dots^{\circ}$

Practice

A-Level

C1
A buoy drifts at constant velocity. At $10 : 00$ it is at $(- 3 i + 5 j)$ km and at $10 : 30$ at $(4 i - 19 j)$ km ( $i$ east, $j$ north). Find its velocity, its speed and its bearing of drift, to $1$ d.p.
[4 marks]
Answervelocity $\dots\dots$ speed $\dots\dots$ bearing $\dots \dots^{\circ}$

C2
A $0.4$ kg particle has velocity $(3 i - 2 j)$ m s $^{- 1}$ at $t = 0$ and $(11 i + 4 j)$ m s $^{- 1}$ at $t = 2$ s. Using $a = \frac{v - u}{t}$ and $F = m a$ , find $a$ , $∣ F ∣$ , and the angle between $F$ and $i$ to $1$ d.p.
[4 marks]
Answer $a = \dots\dots$ $∣ F ∣ = \dots\dots$ N angle $\dots \dots^{\circ}$
C3
$O A B C$ is a quadrilateral with $O A = a$ , $O B = b$ and $O C = c$ . $P$ , $Q$ , $R$ and $S$ are the midpoints of $O A$ , $A B$ , $B C$ and $C O$ . Prove that $P QR S$ is a parallelogram.
[4 marks]

Exam-style questions

Edexcel exam style

D1
Two forces $F_{1} = (2 i - 5 j)$ N and $F_{2} = (p i + q j)$ N act on a particle. Their resultant $R$ acts in the direction of the vector $3 i + j$ .
(a)
Find the angle between $R$ and $i$ , to $1$ d.p.
[2 marks]
Answer $\dots \dots^{\circ}$
(b)
Show that $p - 3 q = - 17$ .
[3 marks]
(c)
Given that $q = 7$ , hence find $p$ and the magnitude of $R$ , in simplified surd form.
[3 marks]
Answer $p = \dots\dots$ $∣ R ∣ = \dots\dots$ N
Total for D1: 8 marks

D2
$O A = a$ and $O B = b$ . $M$ is on $O A$ with $O M : M A = 2 : 1$ , $N$ is the midpoint of $A B$ , and $C$ is on $O B$ extended with $O C = 3 b$ . Priya says that $M$ , $N$ and $C$ lie on a straight line.
Priya is wrong.
(a)
Find $M N$ and $M C$ , and show that $M$ , $N$ and $C$ are not collinear.
[3 marks]
(b)
The point $D$ on $O B$ extended has $O D = k b$ , and $M$ , $N$ and $D$ are collinear. Hence find $k$ and the ratio $M N : N D$ .
[3 marks]
Answer $k = \dots\dots$ $M N : N D = \dots\dots$
Total for D2: 6 marks
D3
$O A B C$ is a parallelogram with $O A = a$ and $O C = c$ . $M$ is the point on $A B$ with $A M : M B = 1 : 2$ . The line $O M$ meets the diagonal $A C$ at $X$ .
Not drawn accurately
(a)
Find $O M$ in terms of $a$ and $c$ .
[2 marks]
Answer $O M = \dots\dots$
(b)
By writing $O X$ in two ways, find the ratio $O X : X M$ .
[4 marks]
Answer $O X : X M = \dots\dots$
(c)
Hence find the ratio $A X : X C$ .
[2 marks]
Answer $A X : X C = \dots\dots$
Total for D3: 8 marks
D4
At $12 : 00$ ship $P$ is at $(2 i + 10 j)$ km with constant velocity $(3 i - 4 j)$ km h $^{- 1}$ , and ship $Q$ is at $(- 5 i + 3 j)$ km with constant velocity $(6 i + 3 j)$ km h $^{- 1}$ . $i$ points east and $j$ north.
(a)
Find the speed of $Q$ , to $3$ s.f.
[2 marks]
Answer $\dots\dots km h^{- 1}$
(b)
Show that, $t$ hours after $12 : 00$ , $P Q = (3 t - 7) i + (7 t - 7) j$ .
[3 marks]
(c)
Hence find the time at which $Q$ is due north of $P$ , and the distance between the ships then.
[3 marks]
Answertime $\dots\dots$ distance $\dots\dots$ km
Total for D4: 8 marks

Extension

Stretch

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

E1
In triangle $O A B$ , $O A = a$ and $O B = b$ . A median joins a vertex to the midpoint of the opposite side. Prove that the three medians meet at one point $G$ , and find $O G$ .
[6 marks]
Hint 1 · What to try
Write the median from $A$ and the median from $B$ with their own unknowns, as in Example 2.
Hint 2 · The first line
Compare coefficients to find where these two medians meet.
Hint 3 · The full method
Then show that point lies on the median from $O$ , to the midpoint of $A B$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the position vector of a point dividing a line in a ratio.
I can prove points are collinear.
I can find where two lines meet by comparing coefficients.
I can model journeys, velocities and forces with vectors.

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