A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 12: Vectors · Lesson 1

3D coordinates and vectors in 3D

124 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-3d-coordinates-and-vectors/

Date

Do Now

Quick questions to start.

A1
From Year 1
$a = 3 i - 4 j$ . Find $∣ a ∣$ and a unit vector in the direction of $a$ .
[3 marks]
Answer
A2
Needed today
$P$ is the point $(2, - 1)$ and $Q$ is the point $(7, 11)$ . Find $P Q$ in terms of $i$ and $j$ , and the distance $P Q$ .
[3 marks]
Answer
A3
Needed today
A triangle has sides $5$ cm, $7$ cm and $8$ cm. Find the angle opposite the side of length $7$ cm.
[2 marks]
Answer
A4
Common slip
$O A = 4 i + j$ and $O B = i + 5 j$ . Find $A B$ .
[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

Find the distance between the points $(1, - 2, 4)$ and $(3, 4, 1)$ .

Hint The distance from

O

(x, y, z)

x^{2} + y^{2} + z^{2}

, as in the diagram. For two points, use the differences in

x

y

and

z

B1
Find the distance between the points $A (- 1, 4, 2)$ and $B (3, - 4, 3)$ .
[3 marks]
Answer

Example 2

The point $P (3, k, 1)$ is $6$ units from $Q (5, 1, - 3)$ . Find the possible values of $k$ .

Hint Write the distance squared in terms of

k

and set it equal to

36

. Expect two answers.

B2
The point $A (k, 2, - 1)$ is $6$ units from $B (4, - 2, 1)$ . Find the possible values of $k$ .
[4 marks]
Answer

Example 3

$a = 2 i - j + 2 k$ . Find $∣ a ∣$ , a unit vector in the direction of $a$ , and the angle $a$ makes with the positive $x$ -axis.

Hint The angle with the

x

-axis has

cos θ = \frac{x -component}{∣ a ∣}

. Keep the sign of the component.

B3
$b = 4 i + 4 j - 7 k$ . Find $∣ b ∣$ , a unit vector in the direction of $b$ , and the angle $b$ makes with the positive $z$ -axis, to 1 decimal place.
[5 marks]
Answer

Example 4

$A$ , $B$ and $C$ are the points $(1, 2, 0)$ , $(3, 3, 2)$ and $(1, 5, 4)$ . Find angle $B A C$ and the area of triangle $A B C$ .

Hint Find the three side lengths from

A B

A C

and

B C

. Then use the cosine rule and area

= \frac{1}{2} b c sin A

B4
$P$ , $Q$ and $R$ are the points $(2, - 1, 3)$ , $(4, 1, 4)$ and $(- 1, 3, 3)$ . Find angle $QP R$ , to 1 decimal place, and the area of triangle $P QR$ , to 3 significant figures.
[6 marks]
Answer

Example 5

$a = 2 i - 3 j + k$ and $b = 6 i + p j + q k$ are parallel. Find $p$ and $q$ .

Hint Parallel means

b = λ a

for one number

λ

. The

i

components give

λ

B5
$u = - 4 i + 6 j + p k$ and $v = q i - 9 j + 3 k$ are parallel. Find $p$ and $q$ .
[4 marks]
Answer

Practice

A-Level

C1
Lena finds a unit vector in the direction of $6 i - 2 j + 3 k$ . She writes: " $6 + 2 + 3 = 11$ , so the unit vector is $\frac{1}{11} (6 i - 2 j + 3 k)$ ". Find her mistake and give the correct unit vector.
[4 marks]
Answer

C2
$O A = 3 i + 2 j - k$ and $O B = 5 i - j + 3 k$ . Tom writes " $A B = O A - O B = - 2 i + 3 j - 4 k$ ". Find his mistake, give the correct $A B$ , and find the angle $A B$ makes with the positive $y$ -axis, to 1 decimal place.
[5 marks]
Answer
C3
The vector $a = t i + 2 j - 2 t k$ has magnitude $7$ . Find the possible values of $t$ . For the positive value, find the angle $a$ makes with the positive $x$ -axis, to 1 decimal place.
[5 marks]
Answer
C4
The vector $2 i + 3 j + 6 k$ makes angles $α$ , $β$ and $γ$ with the positive $x$ -, $y$ - and $z$ -axes. Show that $cos^{2} α + cos^{2} β + cos^{2} γ = 1$ , and find $γ$ to 1 decimal place.
[4 marks]
C5
$A$ , $B$ and $C$ are the points $(1, 1, 1)$ , $(3, 3, 2)$ and $(2, - 1, 3)$ . Show that triangle $A B C$ is right-angled and isosceles, and find its area.
[5 marks]
Answer
C6
Josh says that $a = 2 i - 4 j + 6 k$ and $b = - 3 i + 6 j - 8 k$ are parallel, because the first two components of $b$ are $- \frac{3}{2}$ times those of $a$ . Is he right? Show your working.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The points $A$ , $B$ and $C$ have coordinates $(2, - 1, 3)$ , $(5, 3, 3)$ and $(- 1, 3, 8)$ .
(a)
Find $A B$ and $A C$ .
[2 marks]
Answer
(b)
Find angle $B A C$ , to 1 decimal place.
[4 marks]
Answer
(c)
Hence find the area of triangle $A B C$ , to 3 significant figures.
[2 marks]
Answer
(d)
Find a unit vector in the direction of $B C$ .
[2 marks]
Answer
Total for D1: 10 marks

D2
Kai finds the angle between $a = 4 i - 4 j + 7 k$ and the positive $y$ -axis. He writes: " $∣ a ∣ = 9$ and $cos θ = \frac{4}{9}$ , so $θ = 63. 6^{\circ}$ ".
Kai is wrong.
(a)
Show that $∣ a ∣ = 9$ .
[2 marks]
(b)
Explain Kai's mistake and find the correct angle, to 1 decimal place.
[3 marks]
Answer
(c)
Find the angle between $a$ and the positive $z$ -axis, to 1 decimal place.
[2 marks]
Answer
(d)
The point $P$ is such that $O P$ is in the same direction as $a$ and $O P = 27$ . Find the coordinates of $P$ .
[2 marks]
Answer
Total for D2: 9 marks
D3
The points $P$ and $Q$ lie on the $z$ -axis, and each is $7$ units from the point $A (2, 3, 1)$ . Find the exact area of triangle $A P Q$ .
[8 marks]
Answer
D4
$O A = 2 i + 5 j - k$ and $O B = 6 i + j + 7 k$ . $M$ is the midpoint of $A B$ .
(a)
Find $A B$ and the exact length $A B$ .
[3 marks]
Answer
(b)
Using your answer to (a), find the position vector of $M$ .
[2 marks]
Answer
(c)
Hence find $O M$ and the angle $O M$ makes with the positive $x$ -axis, to 1 decimal place.
[3 marks]
Answer
(d)
The point $N$ lies on $A B$ with $A N : N B = 1 : 3$ . Find the coordinates of $N$ .
[2 marks]
Answer
Total for D4: 10 marks
D5
$a = 3 i + 2 j - k$ and $b = - i + 4 j + 2 k$ .
(a)
Find the values of $p$ and $q$ such that $p a + q b = 11 i - 2 j - 7 k$ .
[4 marks]
Answer
(b)
Find the value of $r$ for which $i + 10 j + r k$ can be written as $p a + q b$ .
[3 marks]
Answer
(c)
Hence find the magnitude of $a + 2 b$ , to 3 significant figures.
[2 marks]
Answer
Total for D5: 9 marks
D6
The diagram shows a cuboid $O A B C D E F G$ . $O$ is the origin, $A$ lies on the $x$ -axis, $C$ on the $y$ -axis and $D$ on the $z$ -axis. $O A = 4$ , $O C = 6$ and $O D = 3$ .
(a)
Write down the coordinates of $F$ and $G$ .
[2 marks]
Answer
(b)
Find the exact length of the diagonal $O F$ .
[2 marks]
Answer
(c)
Find the angle between $O F$ and the $z$ -axis, to 1 decimal place.
[2 marks]
Answer
(d)
The diagonals $O F$ and $A G$ meet at $M$ . Find the acute angle between them, to 1 decimal place.
[4 marks]
Answer
Total for D6: 10 marks

Extension

Stretch

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

E1
The point $P$ lies on the line through $O$ in the direction of $2 i + j + 2 k$ . The point $A$ is $(5, 0, 4)$ . Find the coordinates of the point $P$ that is nearest to $A$ , the least distance $A P$ , and the exact area of triangle $O A P$ for this $P$ .
[9 marks]
Hint 1 · What to try
Any point on the line is $(2 t, t, 2 t)$ for some number $t$ .
Hint 2 · The first line
Write $A P^{2}$ as a quadratic in $t$ . Its least value gives the nearest point.
Hint 3 · The full method
At the nearest point, $A P$ is at right angles to $O P$ , so $A P$ is the height of the triangle on the base $O P$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the distance between two points in three dimensions.
I can find the magnitude of a 3D vector, a unit vector in its direction and the angle it makes with an axis.
I can find the sides, angles and area of a triangle from the coordinates of its corners.
I can use parallel vectors and given lengths to find unknown values.

Answers and mark scheme

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

Get the answers