A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 12: Vectors · Lesson 2

Solving geometric problems with vectors

114 marks

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Quick questions to start.

A1
Last lesson
Find the distance between the points $(1, 2, - 2)$ and $(3, - 4, 1)$ .
[2 marks]
Answer
A2
Needed today
$O A = 2 i + j - k$ and $O B = 4 i + 7 j + 3 k$ . Find the position vector of the midpoint of $A B$ .
[2 marks]
Answer
A3
Common slip
$O A = a$ and $O B = b$ . The point $P$ lies on $A B$ with $A P : P B = 1 : 2$ . Write $O P$ in terms of $a$ and $b$ .
[2 marks]
Answer
A4
From Year 1
$a$ and $b$ are not parallel. $(p + 2) a + 3 b = 5 a + (q - 1) b$ . Find $p$ and $q$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Show that the points $A (1, 2, 3)$ , $B (3, 5, 4)$ and $C (7, 11, 6)$ are collinear.

Hint Find two vectors that share a point, such as

A B

and

A C

. If one is a multiple of the other, the lines are parallel, and the shared point puts all three on one line.

B1
Show that $P (2, - 1, 4)$ , $Q (5, 1, 2)$ and $R (11, 5, - 2)$ are collinear, and find the ratio $P Q : QR$ .
[4 marks]
Answer

Example 2

$A B C D$ is a parallelogram. $A$ , $B$ and $C$ are the points $(1, 0, 2)$ , $(4, 1, 3)$ and $(6, 4, 1)$ . Find the coordinates of $D$ .

Hint Going round

A B C D

in order,

A D = B C

. Take care with the order of the letters.

B2
$A B C D$ is a parallelogram with $A (2, - 1, 1)$ , $B (5, 3, 0)$ and $C (4, 6, - 2)$ . Find the coordinates of $D$ and the exact length of the diagonal $B D$ .
[4 marks]
Answer

Example 3

$A$ is $(1, 4, - 2)$ and $B$ is $(7, 1, 4)$ . The point $P$ lies on $A B$ with $A P : P B = 1 : 2$ . Find the coordinates of $P$ .

Hint

A P : P B = 1 : 2

means

P

\frac{1}{3}

of the way from

A

B

, so

O P = O A + \frac{1}{3} A B

B3
$A$ is $(2, 5, - 1)$ and $B$ is $(12, - 5, 9)$ . The point $P$ lies on $A B$ with $A P : P B = 3 : 2$ . Find the coordinates of $P$ and the exact distance $O P$ .
[4 marks]
Answer

Example 4

In triangle $O A B$ , $O A = a$ and $O B = b$ . $M$ is the midpoint of $A B$ and $N$ lies on $O B$ with $O N : N B = 1 : 2$ . The lines $O M$ and $A N$ meet at $X$ . Find $O X$ .

Hint Write

O X

in two ways: as

λ O M

, and as

O A + μ A N

. As

a

and

b

are not parallel, the coefficients of

a

must match and so must the coefficients of

b

B4
In triangle $O A B$ , $O A = a$ and $O B = b$ . $P$ lies on $O A$ with $O P : P A = 2 : 1$ , and $Q$ is the midpoint of $A B$ . The lines $O Q$ and $B P$ meet at $R$ . Find $O R$ in terms of $a$ and $b$ , and the ratio $O R : R Q$ .
[6 marks]
Answer

Example 5

$a$ , $b$ and $c$ are not coplanar. $(2 p + q) a + (q - r) b$ $+ (p + r) c$ $= 7 a + b + 4 c$ . Find $p$ , $q$ and $r$ .

Hint Three vectors that are not in one plane cannot be combined to give zero unless every coefficient is zero. So the coefficients of

a

, of

b

and of

c

match separately.

B5
$a$ , $b$ and $c$ are not coplanar. $(p + q) a + (2 q - r) b$ $+ (p + 2 r) c$ $= 5 a + 4 b + 9 c$ . Find $p$ , $q$ and $r$ .
[4 marks]
Answer

Practice

A-Level

C1
$A$ , $B$ , $C$ and $D$ are the points $(1, 0, 2)$ , $(3, 1, 5)$ , $(4, 4, 4)$ and $(8, 6, 10)$ . Amir writes: " $A B = 2 i + j + 3 k$ and $C D = 4 i + 2 j + 6 k = 2 A B$ , so $A$ , $B$ , $C$ and $D$ lie on one straight line." Find his mistake and show that $A$ , $B$ and $C$ are not collinear.
[4 marks]
Answer

C2
$A$ is $(4, - 2, 7)$ and $B$ is $(12, 6, - 1)$ . $P$ lies on $A B$ with $A P : P B = 1 : 3$ . Beth writes $O P = O A + \frac{1}{3} A B$ . Explain her mistake and find the coordinates of $P$ .
[4 marks]
Answer
C3
$A$ is $(2, 1, - 3)$ , $B$ is $(5, 3, 1)$ and $C$ is $(k, 7, m)$ . Given that $A$ , $B$ and $C$ are collinear, find $k$ and $m$ .
[4 marks]
Answer
C4
$A (1, 2, 1)$ , $B (3, - 1, 4)$ and $C (6, 0, 2)$ are three corners of the parallelogram $A B C D$ . Sam finds $D$ from $O D = O B + O C - O A$ . Explain his mistake, find the coordinates of $D$ , and check your answer using the midpoints of the diagonals.
[5 marks]
Answer
C5
$O$ is the origin and $A$ , $B$ and $C$ are the points $(2, - 1, 3)$ , $(5, 2, 6)$ and $(1, 4, 0)$ . Show that $O A B C$ is a trapezium, and find the ratio $O A : C B$ .
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The points $A$ and $B$ have coordinates $(1, 3, - 2)$ and $(5, - 1, 6)$ .
(a)
Find the coordinates of the point $P$ on $A B$ with $A P : P B = 1 : 3$ .
[2 marks]
Answer
(b)
The point $C$ is $(8, - 4, 12)$ . Show that $A$ , $B$ and $C$ are collinear, and find the ratio $A B : B C$ .
[4 marks]
Answer
(c)
Find the exact distance $P C$ .
[2 marks]
Answer
(d)
Find the coordinates of the point $E$ such that $B$ is the midpoint of $A E$ .
[2 marks]
Answer
Total for D1: 10 marks

D2
$P$ , $Q$ , $R$ and $S$ are the points $(1, - 2, 3)$ , $(4, 2, 1)$ , $(10, 10, - 3)$ and $(7, 6, - 1)$ . Mia says that $P QR S$ is a parallelogram, because $P Q$ is parallel to $R S$ and $QR$ is parallel to $S P$ .
Mia is wrong.
(a)
Find $P Q$ and $R S$ , and show that they are parallel.
[2 marks]
Answer
(b)
Show that $P$ , $Q$ and $R$ lie on one straight line.
[3 marks]
(c)
Explain why $P QR S$ is not a parallelogram.
[2 marks]
(d)
Show that $S$ lies on $P R$ and find the ratio $P S : S R$ .
[2 marks]
Answer
Total for D2: 9 marks
D3
In triangle $O A B$ , $O A = a$ and $O B = b$ . $M$ is the midpoint of $O A$ , and $P$ lies on $A B$ with $A P : P B = 2 : 1$ . The lines $O P$ and $B M$ meet at $X$ .
(a)
Find $O P$ in terms of $a$ and $b$ .
[2 marks]
Answer
(b)
Find $B M$ in terms of $a$ and $b$ .
[2 marks]
Answer
(c)
Using your answers to (a) and (b), find $O X$ in terms of $a$ and $b$ .
[4 marks]
Answer
(d)
Hence find the ratios $O X : X P$ and $B X : X M$ .
[2 marks]
Answer
Total for D3: 10 marks
D4
$A$ , $B$ and $C$ are the points $(2, 1, - 1)$ , $(6, 3, 5)$ and $(5, - 2, 4)$ . $A B C D$ is a parallelogram whose diagonals meet at $M$ . Find the coordinates of $D$ and of $M$ , find the exact length of the shorter diagonal, and show that $A B C D$ is not a rhombus.
[8 marks]
Answer
D5
$O$ is the origin and $A$ , $B$ and $C$ are the points $(4, 2, - 2)$ , $(3, 6, 2)$ and $(1, 5, 3)$ .
(a)
Show that $O A B C$ is a trapezium.
[3 marks]
(b)
The diagonals $O B$ and $A C$ meet at $X$ . Find the coordinates of $X$ .
[4 marks]
Answer
(c)
Hence find the ratio $O X : X B$ .
[2 marks]
Answer
Total for D5: 9 marks
D6
$O A B C D E F G$ is a cuboid. $O A B C$ is the base, and $D$ , $E$ , $F$ and $G$ are directly above $O$ , $A$ , $B$ and $C$ . $O A = a$ , $O C = c$ and $O D = d$ . Prove, using vectors, that the diagonals $O F$ and $A G$ meet, and that they meet at the midpoint of each.
[7 marks]

Extension

Stretch

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

E1
$O A B C$ is a parallelogram with $O A = a$ and $O C = c$ . $M$ is the midpoint of $A B$ , and $N$ lies on $B C$ with $B N : N C = 1 : 2$ . The lines $O M$ and $A N$ meet at $X$ . Find the ratios $O X : X M$ and $A X : X N$ .
[9 marks]
Hint 1 · What to try
$O B = a + c$ . Write $O M$ and $O N$ in terms of $a$ and $c$ .
Hint 2 · The first line
$O X = λ O M$ and also $O X = O A + μ A N$ .
Hint 3 · The full method
$a$ and $c$ are not parallel, so compare the coefficients of each. $λ$ and $μ$ give the two ratios.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can show that three points are collinear and find the ratio in which one divides the other two.
I can find the fourth vertex of a parallelogram and show that a shape is a trapezium.
I can find the position vector of a point that divides a line in a given ratio.
I can find where two lines meet by comparing coefficients of vectors that are not parallel.

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