GCSE Maths · Higher · Chapter 25: Vectors · Lesson 2

Vectors in geometry

Grades 7–9About 50 minutesNo calculator38 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-vectors-in-geometry/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $(- 6 1) + (65)$ .
[1 mark]
Answer
A2
From chapter 19
P(late) $= 0.1$ each day, independently. Find P(not late on two days).
[1 mark]
Answer
A3
From chapter 3
A pie chart shows 90 people. A sector has angle $6 0^{\circ}$ . How many people is that?
[1 mark]
Answer
A4
From chapter 10
Find the gradient of the line through $(- 2, 0)$ and $(0, - 2)$ .
[1 mark]
Answer

Grades 7–8

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

Example 1

In triangle $O A B$ , $O A = 5 a$ and $O B = 10 b$ . $P$ is on $A B$ with $A P : P B = 2 : 3$ . Write $O P$ in terms of $a$ and $b$ .

Hint Find

A B

first.

A P

\frac{2}{5}

A B

. Then go from

O

A

and on to

P

B1
In triangle $O A B$ , $O A = 15 a$ and $O B = 10 b$ . $P$ is on $A B$ with $A P : P B = 4 : 1$ . Write $O P$ in terms of $a$ and $b$ .
[3 marks]
Answer

Example 2

$O A = 4 a + b$ , $O B = 6 a + 5 b$ and $O C = 9 a + 11 b$ . Show that $A$ , $B$ and $C$ lie on a straight line.

Hint Find

A B

and

B C

. If one is a multiple of the other, they are parallel, and they share the point

B

B2
$O P = 3 a - 2 b$ , $O Q = a + 4 b$ and $O R = - 4 a + 19 b$ . Show that $P$ , $Q$ and $R$ lie on a straight line, and write down the ratio $P Q : QR$ .
[3 marks]

Grades 7–8

C1
$O A = 4 a - 2 b$ and $O B = 6 a + 8 b$ . $M$ is the midpoint of $A B$ . Write $O M$ in terms of $a$ and $b$ .
[2 marks]
Answer
C2
$O A = 3 a$ and $O B = 2 a + 4 b$ . $B$ is the midpoint of $A C$ . Write $O C$ in terms of $a$ and $b$ .
[2 marks]
Answer

C3
$A B C D$ is a parallelogram. $A B = 5 p$ and $A D = 2 q$ . $E$ is the point on $D C$ with $D E : E C = 2 : 3$ . Write $B E$ in terms of $p$ and $q$ .
[3 marks]
Answer
C4
$a$ and $b$ are not parallel. $P Q = 3 a - 2 b$ and $R S = 12 a + k b$ . $P Q$ is parallel to $R S$ . Work out the value of $k$ .
[2 marks]
Answer
C5
$O A = 6 a$ , $O B = 4 a + 4 b$ and $O C = 4 b$ . Show that $O A B C$ is a trapezium, and write down the ratio $C B : O A$ .
[3 marks]
C6
$A B C D E F$ is a regular hexagon. $A B = p$ and $A F = q$ . Write $B D$ in terms of $p$ and $q$ .
[3 marks]
Answer

Grades 8–9

D1
In triangle $O A B$ , $O A = 9 a$ and $O B = 9 b$ . $P$ is on $A B$ with $A P : P B = 4 : 5$ . Lily says, " $O P = \frac{4}{9} O A + \frac{5}{9} O B = 4 a + 5 b$ ."
Show that Lily is wrong, and work out the correct $O P$ .
[3 marks]
Answer

Grade 9 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find a point that divides a line in a given ratio.
I can prove that three points lie on a straight line.
I can use vectors to show that lines in a shape are parallel.

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.