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

Vectors and representing vectors

70 marks

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Date

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Quick questions to start.

A1
Needed today
Simplify $3 (2 x - y) - 2 (x - 4 y)$ .
[2 marks]
Answer
A2
From chapter 3
Solve the simultaneous equations $2 p + q = 7$ and $p - 3 q = 14$ .
[2 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
A3
From chapter 5
The line through $(1, 2)$ and $(5, k)$ is parallel to the line $y = 3 x + 1$ . Find $k$ .
[2 marks]
Answer $k = \dots\dots$
A4
Needed today
The point $P$ divides the line $A B$ in the ratio $A P : P B = 3 : 2$ . $A B = 15$ cm. Find $A P$ .
[2 marks]
Answer $A P = \dots\dots$ cm

Example, then your turn

A-Level

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

Example 1

In the diagram, $O A = a$ , $O B = b$ , $A C = c$ and $B D = d$ . Find, in terms of $a$ , $b$ , $c$ and $d$ , (a) $A B$ (b) $B C$ (c) $C D$ .

Not drawn accurately

Hint Go along vectors you know. Going against an arrow gives a minus sign:

A O = - a

B1
Using the vectors marked in the diagram, find (a) $C A$ (b) $B C$ (c) $D B$ .
[3 marks]
Not drawn accurately
Answer(a) $\dots\dots$ (b) $\dots\dots$ (c) $\dots\dots$

Example 2

$a = 4 i + 2 j$ and $b = - 3 i + 6 j$ . (a) Find $3 a + 2 b$ in $i$ , $j$ form. (b) Find $a - 2 b$ as a column vector. (c) Show that $3 a + 2 b$ is parallel to $i + 3 j$ . (d) Find $λ$ if $a + λ b$ is parallel to $j$ .

Hint Work on the

i

parts and the

j

parts separately. Two vectors are parallel when one is a number times the other. A vector parallel to

j

has no

i

part.

B2
$c = 5 i - 3 j$ and $d = - 3 i + j$ . (a) Find $2 c + 3 d$ in $i$ , $j$ form. (b) Find $c - 4 d$ as a column vector. (c) Show that $2 c + 3 d$ is parallel to $- 2 i + 6 j$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

$O A B$ is a triangle with $O A = a$ and $O B = b$ . $M$ is the midpoint of $O A$ and $N$ is the point on $A B$ with $A N : N B = 1 : 3$ . Find, in terms of $a$ and $b$ , (a) $A B$ (b) $A N$ (c) $O N$ (d) $M N$ .

Not drawn accurately

Hint

A N

\frac{1}{4}

A B

, so

A N = \frac{1}{4} A B

. Then go from

O

A

N

B3
$A B C D$ is a parallelogram with $A B = p$ and $A D = q$ . $M$ is the midpoint of $C D$ and $N$ is the point on $A C$ with $A N : N C = 1 : 3$ . Find, in terms of $p$ and $q$ , (a) $A C$ (b) $A M$ (c) $N M$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$ (c) $\dots\dots$

Practice

A-Level

C1
State, with a reason, whether each vector is parallel to $2 a - 5 b$ . (i) $6 a - 15 b$ (ii) $5 b - 2 a$ (iii) $4 a - 5 b$ (iv) $- a + \frac{5}{2} b$
[4 marks]
Answer(i) $\dots$ (ii) $\dots$ (iii) $\dots$ (iv) $\dots$

C2
The vectors $3 a + k b$ and $4 a - 6 b$ are parallel. Find $k$ .
[2 marks]
Answer $k = \dots\dots$
C3
The points $A$ , $B$ and $C$ are shown on a grid of unit squares. (a) Write $A B$ and $B C$ as column vectors. (b) Find $A B + 2 B C$ in $i$ , $j$ form.
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Exam-style questions

Edexcel exam style

D1
Lily says: "The vectors $6 i - 8 j$ and $- 9 i + 12 j$ are not parallel, because their $i$ parts have different signs."
Lily is wrong.
(a)
Show that the two vectors are parallel, and say whether they point the same way or opposite ways.
[3 marks]
Answer $\dots\dots$
(b)
The vector $(p + 2) i + 3 j$ is also parallel to $6 i - 8 j$ . Find the value of $p$ .
[3 marks]
Answer $p = \dots\dots$
Total for D1: 6 marks

D2
$O A B$ is a triangle with $O A = a$ and $O B = b$ . $P$ is the point on $O A$ with $O P : P A = 2 : 1$ , $Q$ is the point on $O B$ with $O Q : QB = 2 : 1$ , and $R$ is the point on $A B$ with $A R : R B = 1 : 2$ .
Not drawn accurately
(a)
Find $A B$ in terms of $a$ and $b$ .
[1 mark]
Answer $A B = \dots\dots$
(b)
Find $P Q$ in terms of $a$ and $b$ .
[2 marks]
Answer $P Q = \dots\dots$
(c)
Hence show that $P Q$ is parallel to $A B$ , and find the ratio $P Q : A B$ .
[3 marks]
Answer $P Q : A B = \dots\dots$
(d)
Show that $P R$ is parallel to $O B$ .
[3 marks]
Total for D2: 9 marks
D3
$a = 7 i + j$ and $b = p i + p j$ , where $p$ is a constant. The resultant of $a$ and $b$ is parallel to $2 i + j$ .
(a)
Find the value of $p$ .
[3 marks]
Answer $p = \dots\dots$
(b)
Hence find the resultant of $a$ and $b$ .
[1 mark]
Answer $\dots\dots$
(c)
Find the value of $μ$ for which $a - μ b$ is parallel to $j$ .
[3 marks]
Answer $μ = \dots\dots$
(d)
Given that $λ a + ν b = 9 i - 3 j$ , find $λ$ and $ν$ .
[4 marks]
Answer $λ = \dots\dots$ $ν = \dots\dots$
Total for D3: 11 marks
D4
$O A B C$ is a quadrilateral with $O A = 3 a$ , $O C = 2 c$ and $O B = 3 a + 4 c$ , where $a$ and $c$ are not parallel.
(a)
Find $A B$ and $C B$ in terms of $a$ and $c$ .
[2 marks]
Answer $A B = \dots\dots$ $C B = \dots\dots$
(b)
Hence show that $A B$ is parallel to $O C$ , find the ratio $A B : O C$ , and state, with a reason, the type of quadrilateral $O A B C$ is.
[4 marks]
Answer $A B : O C = \dots\dots$ type: $\dots\dots$
(c)
$M$ is the midpoint of $A B$ . Show that $O A M C$ is a parallelogram.
[3 marks]
Total for D4: 9 marks

Extension

Stretch

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

E1
$a$ and $b$ are not parallel. Find every value of $k$ for which $(k + 1) a + 4 b$ is parallel to $a + (k - 2) b$ . For each value, say whether the two vectors point the same way or opposite ways.
[6 marks]
Hint 1 · What to try
Parallel: $(k + 1) a + 4 b = λ (a + (k - 2) b)$ .
Hint 2 · The first line
Compare the $a$ parts and the $b$ parts: $k + 1 = λ$ and $4 = λ (k - 2)$ .
Hint 3 · The full method
So $(k + 1) (k - 2) = 4$ . The sign of $λ$ gives the direction.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can add vectors with the triangle law.
I can find vectors in a triangle or parallelogram.
I can use $i$ , $j$ and column vectors.
I can show vectors are parallel and find an unknown.

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