Figure — understand the maths, don't just answer it

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Vector notation and representation

7 questions

LessonNot started

A particle moves under the action of two forces,

\mathbf{F}_1

and

\mathbf{F}_2

, only.

It is given that

\mathbf{F}_1 = (6\mathbf{i} - 9\mathbf{j})\ \text{N} \qquad \mathbf{F}_2 = (k\mathbf{i} + 4k\mathbf{j})\ \text{N}

where

k

is a constant.

The acceleration of the particle is

(2\mathbf{i} + 3\mathbf{j})\ \text{m s}^{-2}

Find

k

(4 marks)

●●●●●Level 34 marksStart

A particle, of mass

4

kilograms, moves under the action of two forces,

\mathbf{F}_1

and

\mathbf{F}_2

, only.

It is given that

\mathbf{F}_1 = \begin{bmatrix} a \\ 6 \end{bmatrix} \text{newtons} \qquad \mathbf{F}_2 = \begin{bmatrix} 5 \\ b \end{bmatrix} \text{newtons}

where

a

and

b

are constants.

The acceleration of the particle is

\begin{bmatrix} -b \\ a - 2 \end{bmatrix}\ \text{m s}^{-2}

Find the value of

a

and the value of

b

(4 marks)

●●●●●Level 34 marksStart

Find the value of

t

for which

6\mathbf{i} + t\mathbf{j} + 9\mathbf{k}

is parallel to

2\mathbf{i} - \mathbf{j} + 3\mathbf{k}

.

●●●●●Level 32 marksStart

Given

\mathbf{a} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k}

and

\mathbf{b} = 5\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}

, find

3\mathbf{a} - 2\mathbf{b}

and the vector

\mathbf{c}

for which

\mathbf{a} + 2\mathbf{c} = \mathbf{b}

.

●●●●●Level 34 marksStart

Magnitude and direction of a vector

40 questions

LessonNot started

Relative to a fixed origin

O

the point $A$ has position vector $\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$
the point $B$ has position vector $2\mathbf{i} + c\mathbf{j} + 4\mathbf{k}$

where

c

is a constant.
(a) Show that

\left|\overrightarrow{OA}\right| = \sqrt{14}

(1 mark)

Given that

\left|\overrightarrow{OB}\right| = 2\left|\overrightarrow{OA}\right|

(b) find the possible values of

c

.

(3 marks)

●●●●●Level 34 marksStart

The vector

\begin{pmatrix} a \\ b \end{pmatrix}

has magnitude

10\sqrt{2}

and direction

30°

above the positive

x

-axis.

Determine the exact values of

a

and

b

.

(4 marks)

●●●●●Level 34 marksStart

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$ .]

Two remote-controlled model yachts,

P

and

Q

, are sailing on a large lake. Each yacht is moving with constant velocity and the yachts are modelled as particles.

At time

t

seconds, the position vector of

P

is

\mathbf{r}

metres.

When $t = 4$ , $\mathbf{r} = (-3\mathbf{i} + 14\mathbf{j})$
When $t = 10$ , $\mathbf{r} = (15\mathbf{i} + 2\mathbf{j})$

(a) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the velocity of

P

.

(2 marks)
(b) Find an expression for

\mathbf{r}

in terms of

t

,

\mathbf{i}

and

\mathbf{j}

.

(2 marks)

At time

t

seconds, the position vector of

Q

is

\mathbf{s}

metres, where

\mathbf{s} = 8\mathbf{i} - 5\mathbf{j} + t(-\mathbf{i} + 4\mathbf{j})

(c) Find, to the nearest degree, the bearing of

Q

from

P

when

t = 10

(3 marks)

●●●●●Level 37 marksStart

Figure 1 shows a sketch of triangle

ABC

(not accurately drawn). Given that

\vec{AB} = -3\mathbf{i} - 4\mathbf{j} - 5\mathbf{k}

and

\vec{BC} = \mathbf{i} + \mathbf{j} + 4\mathbf{k}

, (a) find

\vec{AC}

, (b) show that

\cos(\angle ABC) = \dfrac{9}{10}

.

●●●●●Level 35 marksStart

The vector

\mathbf{v} = 5\mathbf{i} - 12\mathbf{j}

.
(a) Find

|\mathbf{v}|

. (2 marks)
(b) Find the unit vector in the direction of

\mathbf{v}

. (2 marks)
(c) Find the angle, to 1 d.p., that

\mathbf{v}

makes with the vector

\mathbf{i}

. (3 marks)

●●●●●Level 37 marksStart

Position vectors

44 questions

LessonNot started

Relative to a fixed origin

O

the point $A$ has position vector $\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$
the point $B$ has position vector $2\mathbf{i} + c\mathbf{j} + 4\mathbf{k}$

where

c

is a constant.
(a) Show that

\left|\overrightarrow{OA}\right| = \sqrt{14}

(1 mark)

Given that

\left|\overrightarrow{OB}\right| = 2\left|\overrightarrow{OA}\right|

(b) find the possible values of

c

.

(3 marks)

●●●●●Level 34 marksStart

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$ .]

Two remote-controlled model yachts,

P

and

Q

, are sailing on a large lake. Each yacht is moving with constant velocity and the yachts are modelled as particles.

At time

t

seconds, the position vector of

P

is

\mathbf{r}

metres.

When $t = 4$ , $\mathbf{r} = (-3\mathbf{i} + 14\mathbf{j})$
When $t = 10$ , $\mathbf{r} = (15\mathbf{i} + 2\mathbf{j})$

(a) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the velocity of

P

.

(2 marks)
(b) Find an expression for

\mathbf{r}

in terms of

t

,

\mathbf{i}

and

\mathbf{j}

.

(2 marks)

At time

t

seconds, the position vector of

Q

is

\mathbf{s}

metres, where

\mathbf{s} = 8\mathbf{i} - 5\mathbf{j} + t(-\mathbf{i} + 4\mathbf{j})

(c) Find, to the nearest degree, the bearing of

Q

from

P

when

t = 10

(3 marks)

●●●●●Level 37 marksStart

Points

A

and

B

have position vectors

2\mathbf{i} - \mathbf{j} + 3\mathbf{k}

and

5\mathbf{i} + 3\mathbf{j} - \mathbf{k}

. Find the exact distance

AB

.

●●●●●Level 33 marksStart

The points

A

and

B

have position vectors

\mathbf{a} = 2\mathbf{i} + \mathbf{j}

and

\mathbf{b} = 8\mathbf{i} + 5\mathbf{j}

.
(a) Find

\overrightarrow{AB}

and its magnitude, giving

|\overrightarrow{AB}|

to 3 s.f. (3 marks)
(b) The point

P

divides

AB

in the ratio

1 : 2

. Find the position vector of

P

. (4 marks)

●●●●●Level 37 marksStart

Solving geometric problems with vectors

29 questions

LessonNot started

The points

A

,

B

and

C

have coordinates

(1, 2)

,

(4, 8)

and

(6, 12)

.
(a) Find

\overrightarrow{AB}

and

\overrightarrow{BC}

. (2 marks)
(b) Show that

A

,

B

and

C

are collinear. (3 marks)
(c) Find the ratio

AB : BC

. (2 marks)

●●●●●Level 37 marksStart

In triangle

OAB

,

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

. The point

M

is the midpoint of

AB

, and

G

lies on

OM

with

OG : GM = 2 : 1

.
(a) Find

\overrightarrow{OM}

in terms of

\mathbf{a}

and

\mathbf{b}

. (3 marks)
(b) Show that

\overrightarrow{OG} = \tfrac13(\mathbf{a} + \mathbf{b})

. (3 marks)

●●●●●Level 36 marksStart

The vectors

\mathbf{a} = 3\mathbf{i} + 2\mathbf{j}

and

\mathbf{b} = 6\mathbf{i} + \lambda\mathbf{j}

are parallel. Find the value of

\lambda

. (3 marks)

●●●●●Level 33 marksStart

Modelling with vectors

4 questions

LessonNot started

A particle moves with constant velocity

(2\mathbf{i} - \mathbf{j} + 2\mathbf{k})

m s

^{-1}

and starts at the point

(\mathbf{i} + 2\mathbf{j} - \mathbf{k})

m. Find its speed and its position vector after

3

seconds.

●●●●●Level 33 marksStart

A particle moves so that its position vector at time

t

seconds is

\mathbf{r} = (3\mathbf{i} + 2\mathbf{j}) + t(4\mathbf{i} - \mathbf{j})

metres.
(a) Find the position of the particle when

t = 2

. (2 marks)
(b) Find the speed of the particle, to 3 s.f. (3 marks)
(c) Find the distance of the particle from the origin when

t = 2

. (2 marks)

●●●●●Level 37 marksStart

Two particles

A

and

B

move with position vectors

\mathbf{r}_A = (\mathbf{i} + 2\mathbf{j}) + t(2\mathbf{i} + \mathbf{j})

and

\mathbf{r}_B = 7\mathbf{i} + t(-\mathbf{i} + 2\mathbf{j})

at time

t

seconds.
(a) Show that the particles collide, and find the time of collision. (4 marks)
(b) Find the position vector at which they collide. (2 marks)

●●●●●Level 36 marksStart