Browse Vectors

$P$ passes through $X$ exactly $1$ second after $R$ passes through $X$
$P$ and $R$ are exactly $50$ metres apart $5$ seconds after $R$ passes through $X$

Show that

P

and

R

move along perpendicular lines.

(5 marks)

●●●●●Level 49 marksStart

A particle,

T

, moves around the edges of a triangle

ABC

, in an anticlockwise direction, at a constant speed, as shown in the diagram below.When

T

moves from

A

B

its velocity is

\left(2\sqrt{3}\,\mathbf{i} + 2\mathbf{j}\right)\ \text{m s}^{-1}

(a) Find the speed of

T

.

(1 mark)
(b) When

T

moves from

B

C

its velocity is

\left(-2\sqrt{3}\,\mathbf{i} + 2\mathbf{j}\right)\ \text{m s}^{-1}

Show that the acute angle

ABC = 60^\circ

(2 marks)
(c) It is given that

ABC

is an equilateral triangle.

T

returns to its initial position after

18

seconds.

Vertex

B

lies at the point with position vector

\begin{bmatrix} 5 \\ 1 \end{bmatrix}

metres.

Find the position vector of

C

.

(3 marks)

●●●●●Level 46 marksStart

A particle is moving such that its position vector,

\mathbf{r}

metres, at time

t

seconds, is given by

\mathbf{r} = \mathrm{e}^{2t}\cos t\,\mathbf{i} + \mathrm{e}^{2t}\sin t\,\mathbf{j}

Show that the magnitude of the acceleration of the particle,

a\ \text{m s}^{-2}

, is given by

a = 5\mathrm{e}^{2t}

Fully justify your answer.

(7 marks)

●●●●●Level 47 marksStart

(80\mathbf{i} + 150\mathbf{j})\ \text{km h}^{-1}

At 14:00, the aircraft is at the point with position vector

(-20\mathbf{i} - 30\mathbf{j})

km.

The aircraft is modelled as a particle moving in a horizontal plane.
(a) Find the speed of the aircraft.

(2 marks)

At time

t

hours after 14:00, the position vector of the aircraft is

\mathbf{a}

km.
(b) Find an expression for

\mathbf{a}

in terms of

\mathbf{i}

\mathbf{j}

and

t

.

(2 marks)

A tall radio mast is modelled as being at the point with position vector

(10\mathbf{i} + 19\mathbf{j})

km in the same horizontal plane.

Aircraft are not permitted to fly within

3

km of the mast.
(c) (i) Find the value of

t

when the aircraft is closest to the mast.

(ii) Hence determine whether the aircraft can continue on its course without breaking this rule.

(7 marks)

●●●●●Level 411 marksStart

A

and

B

, are crossing a large frozen lake. The snowmobiles are modelled as particles.

Snowmobile

A

is moving with constant velocity.

At 10:00, the position vector of

A

(-6\mathbf{i} + 11\mathbf{j})

km.

At 10:40, the position vector of

A

(10\mathbf{i} - \mathbf{j})

km.

At time

t

hours after 10:00, the position vector of

A

\mathbf{r}_A

km.
(a) Show that

\mathbf{r}_A = (-6 + 24t)\mathbf{i} + (11 - 18t)\mathbf{j}

(4 marks)

The speed of

A

V\ \text{m s}^{-1}

(b) Find the value of

V

.

(2 marks)

Snowmobile

B

is moving with constant velocity

(16\mathbf{i} - 20\mathbf{j})\ \text{km h}^{-1}

At 10:00, the position vector of

B

(10\mathbf{i} + 9\mathbf{j})

km.

At 11:15,

B

passes through the point

Q

.
(c) Show that

A

also passes through

Q

.

(5 marks)

●●●●●Level 411 marksStart

Given

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

and

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

, find

|\mathbf{a} - \mathbf{b}|

in surd form.

●●●●●Level 43 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

[In this question, $\mathbf{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north. Position vectors are given relative to a fixed origin $O$ .]

At time

t

seconds,

t \ge 1

, the position vector of a particle

P

\mathbf{r}

metres, where

\mathbf{r} = (2t^3 - 8t^2 + c)\,\mathbf{i} + (t^3 - 2t^2)\,\mathbf{j}

and

c

is a constant.

When

t = 3

, the bearing of

P

from

O

045^\circ

(a) Show that

c = 27

(3 marks)
(b) Find the speed of

P

when

t = 3

(4 marks)

When

t = T

P

is accelerating in the direction of

(2\mathbf{i} + 5\mathbf{j})

.
(c) Find the value of

T

.

(4 marks)

●●●●●Level 511 marksStart

The straight line

l

passes through the points

A

and

B

.

Relative to a fixed origin

O

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

(a) Find

\overrightarrow{AB}

(1 mark)

Given that a point

P

lies on

l

such that

\left|\overrightarrow{AP}\right| = 2\left|\overrightarrow{BP}\right|

(b) find the possible position vectors of

P

.

(4 marks)
(c) State, giving a reason, which of the two possible positions of

P

lies between

A

and

B

.

(1 mark)

●●●●●Level 56 marksStart

Relative to a fixed origin

O

the point $A$ has position vector $\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}$
the point $B$ has position vector $7\mathbf{i} + q\mathbf{j} - 4\mathbf{k}$

where

q

is a constant.
(a) Show that

\left|\overrightarrow{OA}\right| = 9

(1 mark)

Given that

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

(b) find the possible values of

q

.

(2 marks)

For the positive value of

q

, the point

C

is such that

OACB

is a parallelogram.
(c) Find the exact area of

OACB

.

(3 marks)

●●●●●Level 56 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

1.2

kg is moving on a smooth horizontal surface under the action of a single horizontal force

\mathbf{F}

N. At time

t

seconds, where

t \geq 0

, the velocity

\mathbf{v}\ \text{m s}^{-1}

P

, relative to a fixed origin

O

, is given by

\mathbf{v} = (t^2 - 8t + 20)\mathbf{i} + (t - 6)\mathbf{j}.

(a) Show that

P

is never stationary.

(2 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the acceleration of

P

at time

t

.

(1 mark)
(c) Determine the magnitude of

\mathbf{F}

when

P

is moving in the direction of the vector

5\mathbf{i} - \mathbf{j}

. Give your answer correct to 3 significant figures.

(5 marks)

When

t = 3

P

is at the point with position vector

(-9\mathbf{i} + 1.5\mathbf{j})

m.
(d) Determine the bearing of

P

from

O

at time

t = 6

.

(5 marks)

●●●●●Level 513 marksStart

The points

A

B

and

C

have coordinates

(1, 0, 2)

(5, 8, -2)

and

(7, 6, k)

respectively, where

k

is a constant.
(a) Given that angle

ACB

is a right angle, find the possible values of

k

. (3 marks)
(b) Given also that

k > 0

, find the exact shortest distance from

C

to the line

AB

. (3 marks)
(c) The point

D

is the point on

AB

that is closest to

C

. Find the ratio

AD : DB

in its simplest form. (2 marks)

●●●●●Level 58 marksStart

Relative to a fixed origin

O

, the points

A

and

B

have position vectors

3\mathbf{j} - 3\mathbf{k}

and

6\mathbf{i} + 3\mathbf{k}

respectively.

The point

P

lies on the line segment

AB

and is such that

OP

is perpendicular to

AB

.
(a) Find the position vector of

P

. (5 marks)
(b) Hence find the exact area of triangle

OAB

. (2 marks)

●●●●●Level 57 marksStart

Relative to a fixed origin

O

, the points

A

B

and

C

have position vectors

\mathbf{a} = -\mathbf{i} + \mathbf{j} + 4\mathbf{k}

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

\qquad \mathbf{c} = 5\mathbf{i} + 7\mathbf{j} + 4\mathbf{k}

.
(a) Show that angle

BAC = \dfrac{\pi}{4}

. (4 marks)

The point

P

lies on the line segment

AB

and is such that angle

APC = \dfrac{\pi}{2}

.
(b) Find the position vector of

P

. (3 marks)
(c) Find the area of triangle

ABC

. (2 marks)

●●●●●Level 59 marksStart

Relative to a fixed origin

O

, the points

A

B

and

C

have position vectors

\mathbf{a} = \mathbf{i} + 2\mathbf{k}, \qquad \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \qquad \text{and} \qquad \mathbf{c} = 7\mathbf{i} + 6\mathbf{j} - \mathbf{k}.

(a) Show that angle

ABC

is a right angle. (3 marks)
(b) The point

D

lies on

AC

and is such that

BD

is perpendicular to

AC

. Determine the position vector of

D

. (5 marks)
(c) Find the exact length of

BD

. (1 mark)

●●●●●Level 59 marksStart

A

and a lifeboat

B

are each moving with constant velocity.

the velocity of $A$ is $36\ \text{km h}^{-1}$ due east
the velocity of $B$ is $29\ \text{km h}^{-1}$ on a bearing of $\alpha$ , where $\tan\alpha = \dfrac{21}{20}$

The boats are modelled as particles.
(a) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the velocity of

B

\text{km h}^{-1}

(2 marks)

At 13:00

the position vector of $A$ is $(-4\mathbf{i} + 12\mathbf{j})$ km
the position vector of $B$ is $(5\mathbf{i} + 5\mathbf{j})$ km

At time

t

hours after 13:00

the position vector of $A$ is $\mathbf{r}$ km, where $\mathbf{r} = (-4 + 36t)\mathbf{i} + 12\mathbf{j}$
the position vector of $B$ is $\mathbf{s}$ km

(b) Find an expression for

\mathbf{s}

in terms of

t

\mathbf{i}

and

\mathbf{j}

.

(2 marks)
(c) Show that at time

t

hours after 13:00,

\overrightarrow{AB} = [(9 - 15t)\mathbf{i} + (20t - 7)\mathbf{j}]

km

(2 marks)
(d) Show that the boats will never collide.

(3 marks)
(e) Find the distance between the boats when

B

is north-east of

A

.

(4 marks)

●●●●●Level 513 marksStart

[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

Two drones,

A

and

B

, are flying at the same constant height. Each drone is moving with constant acceleration and the drones are modelled as particles.

At time

t

seconds, drone

A

has velocity

\mathbf{v}_A\ \text{m s}^{-1}

, where

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

(a) Find the magnitude of the acceleration of

A

.

(2 marks)

When

t = 1

, the velocity of

B

(5\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-1}

When

t = 4

, the velocity of

B

(-\mathbf{i} + 5\mathbf{j})\ \text{m s}^{-1}

(b) Find the acceleration of

B

, giving your answer in terms of

\mathbf{i}

and

\mathbf{j}

.

(2 marks)
(c) Find the velocity of

B

at time

t = 0

, giving your answer in terms of

\mathbf{i}

and

\mathbf{j}

.

(2 marks)

At time

T_1

seconds, where

T_1 > 0

, both drones are moving with the same speed.
(d) Find the exact value of

T_1

(4 marks)

At time

T_2

seconds, where

T_2 > 0

, both drones are moving in the same direction.
(e) Show that

5T_2^2 + pT_2 + q = 0

, where

p

and

q

are integers to be found.

(3 marks)

●●●●●Level 513 marksStart

Triangle

PQR

has

\vec{PQ} = 2\mathbf{i} + 2\mathbf{j} - \mathbf{k}

and

\vec{QR} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}

. Find the size of angle

PQR

●●●●●Level 53 marksStart

Position vectors

44 questions

LessonNot started

Relative to a fixed origin

O

the point $A$ has position vector $\mathbf{i} - 4\mathbf{j} + 6\mathbf{k}$
the point $B$ has position vector $-3\mathbf{i} + 2\mathbf{j} + a\mathbf{k}$

where

a

is a positive integer.
(a) Show that

\left|\overrightarrow{OA}\right| = \sqrt{53}

(1 mark)
(b) Find the smallest value of

a

for which

\left|\overrightarrow{OB}\right| > \left|\overrightarrow{OA}\right|

(2 marks)

●●●●●Level 23 marksStart

Relative to a fixed origin

O

the point $A$ has position vector $5\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$
the point $B$ has position vector $-\mathbf{i} + 8\mathbf{j} + 2\mathbf{k}$

(a) Find

\overrightarrow{AB}

(1 mark)
(b) Find

\left|\overrightarrow{AB}\right|

, writing your answer as a fully simplified surd.

(2 marks)

●●●●●Level 23 marksStart

The points

A

and

B

have position vectors

\begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}

and

\begin{pmatrix} 6 \\ 1 \\ -2 \end{pmatrix}

respectively.
(a) Find the exact length of

AB

.

(2 marks)
(b) Find the position vector of the midpoint of

AB

.

(1 mark)

●●●●●Level 23 marksStart

Two particles,

P

and

Q

, are moving in separate straight lines across a smooth horizontal surface.

P

moves with constant velocity

(4\mathbf{i} - 6\mathbf{j})\ \text{m s}^{-1}

Q

moves from the point with position vector

(-3\mathbf{i} + 5\mathbf{j})

metres to the point with position vector

(7\mathbf{i} - 10\mathbf{j})

metres.

Show that

P

and

Q

move along parallel lines.

(3 marks)

●●●●●Level 23 marksStart

Relative to an origin

O

, points

A

and

B

have position vectors

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

and

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

. Find

\vec{AB}

●●●●●Level 22 marksStart

The points

A

and

B

have coordinates

(1, 2, -3)

and

(3, 5, 3)

respectively.
(a) Find

\overrightarrow{AB}

. (2 marks)
(b) Find the distance

AB

. (2 marks)

●●●●●Level 24 marksStart

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

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

\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

\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

PQRS

is a parallelogram.

Given that

$\overrightarrow{PQ} = 6\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$
$\overrightarrow{QR} = 2\mathbf{i} + 6\mathbf{j} + 3\mathbf{k}$

(a) show that parallelogram

PQRS

is a rhombus.

(2 marks)
(b) Find the exact area of the rhombus

PQRS

.

(4 marks)

●●●●●Level 46 marksStart

Relative to a fixed origin

O

the point $A$ has position vector $3\mathbf{i} + 2\mathbf{j} - \mathbf{k}$
the point $B$ has position vector $5\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$
the point $C$ has position vector $9\mathbf{i} + p\mathbf{j} + 11\mathbf{k}$

where

p

is a constant.

Given that

A

B

and

C

lie on a straight line,
(a) find the value of

p

.

(3 marks)

The point

D

lies on

OB

extended and

\overrightarrow{CD}

is parallel to

\overrightarrow{OA}

(b) Find

\left|\overrightarrow{OD}\right|

, writing your answer as a fully simplified surd.

(3 marks)

●●●●●Level 46 marksStart

The straight line

l

passes through the points

A

and

B

.

Relative to a fixed origin

O

the point $A$ has position vector $4\mathbf{i} + \mathbf{j} - 2\mathbf{k}$
the point $B$ has position vector $8\mathbf{i} - 7\mathbf{j} + 10\mathbf{k}$

(a) Find

\overrightarrow{AB}

(1 mark)

Given that a point

P

lies on

l

such that

\left|\overrightarrow{AP}\right| = 3\left|\overrightarrow{BP}\right|

(b) find the possible position vectors of

P

.

(4 marks)

●●●●●Level 45 marksStart

KLMN

is a parallelogram.

Given that

$\overrightarrow{KL} = \mathbf{i} + 8\mathbf{j} + 4\mathbf{k}$
$\overrightarrow{LM} = 8\mathbf{i} + t\mathbf{j} + \mathbf{k}$

where

t

is a positive constant,
(a) find the value of

t

for which

KLMN

is a rhombus.

(2 marks)
(b) Using your answer to part (a), find the exact area of the rhombus

KLMN

.

(4 marks)

●●●●●Level 46 marksStart

The point

A

is such that

|\overrightarrow{OA}| = 10

and

\overrightarrow{OA}

makes an angle of

135°

with the positive

x

-axis, measured anticlockwise.
(a) (i) On a sketch of the

x

- and

y

-axes, mark the position of the point

A

.

(ii) Find the position vector of

A

in terms of

\mathbf{i}

and

\mathbf{j}

.

(4 marks)

The point

B

has position vector

8\mathbf{i}

.
(b) Find the exact area of triangle

AOB

.

(2 marks)

OABC

is a parallelogram.
(c) Find the position vector of

C

in terms of

\mathbf{i}

and

\mathbf{j}

.

(2 marks)

●●●●●Level 48 marksStart

ABCD

is a quadrilateral. The points

A

and

B

have position vectors

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

and

-3\mathbf{i}

respectively.
(a) Find the length

AB

.

(1 mark)

The point

C

has position vector

p\mathbf{i} + p\mathbf{j}

, where

p

is a constant greater than

1

. It is given that

BC = AB

.
(b) Determine the position vector of

C

.

(3 marks)

The point

M

is the midpoint of

AC

, and

\overrightarrow{MD} = \overrightarrow{BM}

.
(c) Determine the position vector of

D

.

(2 marks)
(d) State the name of the quadrilateral

ABCD

, giving a reason for your answer.

(2 marks)

●●●●●Level 48 marksStart

The points

O

and

A

have position vectors

\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

and

\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}

respectively. The point

P

is such that

\overrightarrow{OP} = k\overrightarrow{OA}

, where

k

is a positive constant.
(a) Find, in terms of

k

, the length of

OP

.

(1 mark)

The point

B

has position vector

\begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}

and angle

OPB

is a right angle.
(b) Determine the value of

k

.

(4 marks)

●●●●●Level 45 marksStart

The diagram shows triangle

OAB

, where

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

. The point

P

is the midpoint of

OB

, and the point

Q

lies on

AB

such that

AQ : QB = 1 : 3

.(a) Find

\overrightarrow{PQ}

in terms of

\mathbf{a}

and

\mathbf{b}

.

(2 marks)

The line

OA

is produced to the point

R

so that

P

Q

and

R

are collinear.
(b) Explain why

\overrightarrow{PR} = k(3\mathbf{a} - \mathbf{b})

for some constant

k

.

(2 marks)
(c) Hence determine the ratio

OA : AR

.

(4 marks)

●●●●●Level 48 marksStart

The diagram shows a cuboid

OABCDEFG

. The vertices

D

E

F

and

G

lie vertically above

O

A

B

and

C

respectively.

\overrightarrow{OA} = \mathbf{a}

\overrightarrow{OC} = \mathbf{c}

and

\overrightarrow{OD} = \mathbf{d}

The point $M$ lies on $GF$ such that $GM : MF = 1 : 2$ .
The point $X$ lies on $AM$ such that $AX : XM = 3 : 2$ .

(a) Find

\overrightarrow{OX}

in terms of

\mathbf{a}

\mathbf{c}

and

\mathbf{d}

.

(4 marks)
(b) Hence show that the lines

OF

and

AM

intersect.

(2 marks)

●●●●●Level 46 marksStart

The points

A

and

B

have position vectors

\mathbf{a}

and

\mathbf{b}

respectively, relative to the origin

O

.
(a) State, with reference to the points

A

and

B

, the geometrical significance of

(i)

|\mathbf{a} - \mathbf{b}|

,

(ii)

\tfrac{1}{3}(\mathbf{a} + 2\mathbf{b})

.

(2 marks)

The point

P

has position vector

\mathbf{p}

, where

\left|\mathbf{p} - \tfrac{1}{3}(\mathbf{a} + 2\mathbf{b})\right| = \tfrac{1}{3}|\mathbf{a} - \mathbf{b}|

.
(b) (i) Describe the locus of

P

.

(ii) Explain why

B

lies on this locus.

(2 marks)

It is now given that

\mathbf{a} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}

and

\mathbf{b} = \begin{pmatrix} 7 \\ 7 \end{pmatrix}

.
(c) Find a cartesian equation of the locus of

P

.

(4 marks)

●●●●●Level 48 marksStart

OACB

is a parallelogram, with

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

, so that

\overrightarrow{OC} = \mathbf{a} + \mathbf{b}

.(a) Prove that the diagonals

OC

and

AB

bisect each other. (4 marks)
(b) Given that

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

and

\mathbf{b} = 4\mathbf{i} - \mathbf{j}

, find the exact lengths of the two diagonals. (4 marks)

●●●●●Level 48 marksStart

The points

A

B

and

C

have position vectors, relative to the origin

O

\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \qquad \overrightarrow{OB} = 4\mathbf{i} + 6\mathbf{j} + 3\mathbf{k}, \qquad \overrightarrow{OC} = \mathbf{i} + 5\mathbf{j} + 7\mathbf{k}

(a) Find

\overrightarrow{AB}

and

\overrightarrow{AC}

, and show that the triangle

ABC

is isosceles. (4 marks)
(b) Find the exact length of

BC

. (2 marks)
(c) Use the cosine rule to find angle

BAC

, correct to

1

decimal place. (3 marks)

●●●●●Level 49 marksStart

Relative to a fixed origin

O

, the points

A

and

B

have position vectors

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

and

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

respectively.

The point

P

lies on the straight line through

A

and

B

, and is the point on this line that is closest to

O

.
(a) Find the position vector of

P

. (4 marks)
(b) Hence find the exact area of triangle

OAB

. (3 marks)

●●●●●Level 47 marksStart

The points

A

and

B

have position vectors

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

and

\mathbf{b} = 4\mathbf{i} + 3\mathbf{j} - \mathbf{k}

.
(a) Find

\left|\overrightarrow{AB}\right|

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

\overrightarrow{AB}

. (1 mark)
(c) The point

C

has position vector

\mathbf{c} = c\,\mathbf{j} + 5\mathbf{k}

, where

c

is a constant. Given that

\left|\overrightarrow{AC}\right| = \left|\overrightarrow{AB}\right|

, find the possible values of

c

, giving your answers in exact form. (3 marks)
(d) The point

P

lies on

AB

with

AP : PB = 3 : 1

. Find the position vector of

P

. (2 marks)

●●●●●Level 48 marksStart

Relative to a fixed origin

O

, the points

A

and

B

have position vectors

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

and

\mathbf{b} = 6\mathbf{i} + 3\mathbf{j} + \mathbf{k}

.
(a) Find the distance from

O

to the midpoint

M

AB

, giving your answer in the form

p\sqrt{2}

. (2 marks)
(b) The point

C

is such that

OACB

is a parallelogram. Find the position vector of

C

, and show that

|BC| = \sqrt{30}

. (3 marks)
(c) Determine the size of angle

AOB

, giving your answer to 1 decimal place. (4 marks)

●●●●●Level 49 marksStart

Relative to a fixed origin

O

, the points

A

B

and

C

have position vectors

\mathbf{a} = 2\mathbf{i} - \mathbf{j} + 4\mathbf{k}, \qquad \mathbf{b} = 5\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}, \qquad \mathbf{c} = 11\mathbf{i} + 8\mathbf{j} - 14\mathbf{k}

(a) Find

\overrightarrow{AB}

and

\overrightarrow{BC}

. (3 marks)
(b) Show that

A

B

and

C

are collinear, and state the ratio

AB : BC

. (2 marks)
(c) Find the exact length of

AC

. (2 marks)

●●●●●Level 47 marksStart

Two particles,

P

and

Q

, are moving in separate straight lines across a smooth horizontal surface.

P

moves with constant velocity

(8\mathbf{i} - 6\mathbf{j})\ \text{m s}^{-1}

Q

moves from the point with position vector

(-5\mathbf{i} + 9\mathbf{j})

metres to the point with position vector

(15\mathbf{i} - 6\mathbf{j})

metres during a

2.5

second period.
(a) Show that

P

and

Q

move along parallel lines.

(3 marks)
(b) Megan says

“

Q

is also moving with a constant velocity of

(8\mathbf{i} - 6\mathbf{j})\ \text{m s}^{-1}

”

Explain why Megan may be incorrect.

(1 mark)
(c) A third particle,

R

, is moving with a constant speed of

6\ \text{m s}^{-1}

, in a straight line, across the same surface.

P

and

R

move along lines that intersect at a fixed point

X

.

It is given that:

$P$ passes through $X$ exactly $1$ second after $R$ passes through $X$
$P$ and $R$ are exactly $50$ metres apart $5$ seconds after $R$ passes through $X$

Show that

P

and

R

move along perpendicular lines.

(5 marks)

●●●●●Level 49 marksStart

Two particles,

P

and

Q

, move in the same horizontal plane.

Particle

P

is initially at rest at the point with position vector

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

metres and moves with constant acceleration

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

Particle

Q

moves in a straight line, passing through the points with position vectors

(10\mathbf{i} - 3\mathbf{j})

metres and

(\mathbf{i} + c\mathbf{j})

metres, where

c

is a constant.

P

and

Q

are moving along parallel paths.
(a) Show that

c = 3

(4 marks)
(b) (i) Find an expression for the position vector of

P

at time

t

seconds.

(ii) Hence, prove that the paths of

P

and

Q

are not collinear.

(4 marks)

●●●●●Level 48 marksStart

A particle,

T

, moves around the edges of a triangle

ABC

, in an anticlockwise direction, at a constant speed, as shown in the diagram below.When

T

moves from

A

B

its velocity is

\left(2\sqrt{3}\,\mathbf{i} + 2\mathbf{j}\right)\ \text{m s}^{-1}

(a) Find the speed of

T

.

(1 mark)
(b) When

T

moves from

B

C

its velocity is

\left(-2\sqrt{3}\,\mathbf{i} + 2\mathbf{j}\right)\ \text{m s}^{-1}

Show that the acute angle

ABC = 60^\circ

(2 marks)
(c) It is given that

ABC

is an equilateral triangle.

T

returns to its initial position after

18

seconds.

Vertex

B

lies at the point with position vector

\begin{bmatrix} 5 \\ 1 \end{bmatrix}

metres.

Find the position vector of

C

.

(3 marks)

●●●●●Level 46 marksStart

(80\mathbf{i} + 150\mathbf{j})\ \text{km h}^{-1}

At 14:00, the aircraft is at the point with position vector

(-20\mathbf{i} - 30\mathbf{j})

km.

The aircraft is modelled as a particle moving in a horizontal plane.
(a) Find the speed of the aircraft.

(2 marks)

At time

t

hours after 14:00, the position vector of the aircraft is

\mathbf{a}

km.
(b) Find an expression for

\mathbf{a}

in terms of

\mathbf{i}

\mathbf{j}

and

t

.

(2 marks)

A tall radio mast is modelled as being at the point with position vector

(10\mathbf{i} + 19\mathbf{j})

km in the same horizontal plane.

Aircraft are not permitted to fly within

3

km of the mast.
(c) (i) Find the value of

t

when the aircraft is closest to the mast.

(ii) Hence determine whether the aircraft can continue on its course without breaking this rule.

(7 marks)

●●●●●Level 411 marksStart

A

and

B

, are crossing a large frozen lake. The snowmobiles are modelled as particles.

Snowmobile

A

is moving with constant velocity.

At 10:00, the position vector of

A

(-6\mathbf{i} + 11\mathbf{j})

km.

At 10:40, the position vector of

A

(10\mathbf{i} - \mathbf{j})

km.

At time

t

hours after 10:00, the position vector of

A

\mathbf{r}_A

km.
(a) Show that

\mathbf{r}_A = (-6 + 24t)\mathbf{i} + (11 - 18t)\mathbf{j}

(4 marks)

The speed of

A

V\ \text{m s}^{-1}

(b) Find the value of

V

.

(2 marks)

Snowmobile

B

is moving with constant velocity

(16\mathbf{i} - 20\mathbf{j})\ \text{km h}^{-1}

At 10:00, the position vector of

B

(10\mathbf{i} + 9\mathbf{j})

km.

At 11:15,

B

passes through the point

Q

.
(c) Show that

A

also passes through

Q

.

(5 marks)

●●●●●Level 411 marksStart

Points

A

and

B

have position vectors

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

and

5\mathbf{i} + 4\mathbf{j} + 14\mathbf{k}

. Find the position vector of the midpoint

M

AB

, and the distance

AM

●●●●●Level 44 marksStart

Relative to an origin

O

, the points

A

B

and

C

have position vectors

\mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k}, \qquad \mathbf{b} = 3\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \qquad \mathbf{c} = 2\mathbf{i} + 3\mathbf{j}.

(a) Find

\overrightarrow{BA}

and

\overrightarrow{BC}

. (3 marks)
(b) Find angle

ABC

, giving your answer to

1

decimal place. (4 marks)

●●●●●Level 47 marksStart

Relative to an origin

O

, the point

A

has position vector

\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}

and the point

B

has position vector

\mathbf{b} = 6\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}

.
(a) Show that

OA = OB

. (2 marks)
(b) Find

\overrightarrow{AB}

and show that

AB = 7\sqrt{2}

. (3 marks)
(c) Find a unit vector in the direction of

\overrightarrow{AB}

. (2 marks)

●●●●●Level 47 marksStart

The point

P

has position vector

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

, and

\overrightarrow{PQ} = 2\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}

.
(a) Find the position vector of

Q

. (2 marks)
(b) Find

PQ

. (2 marks)
(c) The point

R

lies on

PQ

produced so that

\overrightarrow{PR} = 3\,\overrightarrow{PQ}

. Find the position vector of

R

. (3 marks)

●●●●●Level 47 marksStart

The straight line

l

passes through the points

A

and

B

.

Relative to a fixed origin

O

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

(a) Find

\overrightarrow{AB}

(1 mark)

Given that a point

P

lies on

l

such that

\left|\overrightarrow{AP}\right| = 2\left|\overrightarrow{BP}\right|

(b) find the possible position vectors of

P

.

(4 marks)
(c) State, giving a reason, which of the two possible positions of

P

lies between

A

and

B

.

(1 mark)

●●●●●Level 56 marksStart

Relative to a fixed origin

O

the point $A$ has position vector $\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}$
the point $B$ has position vector $7\mathbf{i} + q\mathbf{j} - 4\mathbf{k}$

where

q

is a constant.
(a) Show that

\left|\overrightarrow{OA}\right| = 9

(1 mark)

Given that

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

(b) find the possible values of

q

.

(2 marks)

For the positive value of

q

, the point

C

is such that

OACB

is a parallelogram.
(c) Find the exact area of

OACB

.

(3 marks)

●●●●●Level 56 marksStart

Relative to a fixed origin

O

, the line

l

has equation

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

where

\lambda

is a scalar parameter.

The point

A

and the point

B

lie on line

l

.

Given that

$A$ has coordinates $(1, a, 2)$
$B$ has coordinates $(b, 6, -2)$

(a) find the value of the constant

a

and the value of the constant

b

.

(2 marks)
(b) Hence show that

\overrightarrow{AB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}

(2 marks)

The point

C

has coordinates

(3, 9, 5)

.
(c) Find the size of angle

BAC

, giving your answer in degrees to one decimal place.

(4 marks)

The point

D

lies on

l

such that the area of triangle

ACD

is three times the area of triangle

ABC

.
(d) Find the coordinates of the two possible positions of

D

.

(4 marks)

●●●●●Level 512 marksStart

The points

A

B

and

C

have coordinates

(1, 0, 2)

(5, 8, -2)

and

(7, 6, k)

respectively, where

k

is a constant.
(a) Given that angle

ACB

is a right angle, find the possible values of

k

. (3 marks)
(b) Given also that

k > 0

, find the exact shortest distance from

C

to the line

AB

. (3 marks)
(c) The point

D

is the point on

AB

that is closest to

C

. Find the ratio

AD : DB

in its simplest form. (2 marks)

●●●●●Level 58 marksStart

Relative to a fixed origin

O

, the points

A

and

B

have position vectors

3\mathbf{j} - 3\mathbf{k}

and

6\mathbf{i} + 3\mathbf{k}

respectively.

The point

P

lies on the line segment

AB

and is such that

OP

is perpendicular to

AB

.
(a) Find the position vector of

P

. (5 marks)
(b) Hence find the exact area of triangle

OAB

. (2 marks)

●●●●●Level 57 marksStart

Relative to a fixed origin

O

, the points

A

B

and

C

have position vectors

\mathbf{a} = -\mathbf{i} + \mathbf{j} + 4\mathbf{k}

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

\qquad \mathbf{c} = 5\mathbf{i} + 7\mathbf{j} + 4\mathbf{k}

.
(a) Show that angle

BAC = \dfrac{\pi}{4}

. (4 marks)

The point

P

lies on the line segment

AB

and is such that angle

APC = \dfrac{\pi}{2}

.
(b) Find the position vector of

P

. (3 marks)
(c) Find the area of triangle

ABC

. (2 marks)

●●●●●Level 59 marksStart

Figure 2 shows the triangle

OAB

, where

\overrightarrow{OA} = \mathbf{a}

and

\overrightarrow{OB} = \mathbf{b}

.

The point

M

is the midpoint of

OB

. The point

N

lies on

AB

with

AN : NB = 1 : 3

. The lines

ON

and

AM

intersect at the point

X

.Figure 2 (not accurately drawn)

Given that

\overrightarrow{OX} = \lambda\,\overrightarrow{ON}

and

\overrightarrow{AX} = \mu\,\overrightarrow{AM}

, where

\lambda

and

\mu

are scalars, find the value of

\lambda

and the value of

\mu

. (7 marks)

●●●●●Level 57 marksStart

Relative to a fixed origin

O

, the points

A

B

and

C

have position vectors

\mathbf{a} = \mathbf{i} + 2\mathbf{k}, \qquad \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \qquad \text{and} \qquad \mathbf{c} = 7\mathbf{i} + 6\mathbf{j} - \mathbf{k}.

(a) Show that angle

ABC

is a right angle. (3 marks)
(b) The point

D

lies on

AC

and is such that

BD

is perpendicular to

AC

. Determine the position vector of

D

. (5 marks)
(c) Find the exact length of

BD

. (1 mark)

●●●●●Level 59 marksStart

A

and a lifeboat

B

are each moving with constant velocity.

the velocity of $A$ is $36\ \text{km h}^{-1}$ due east
the velocity of $B$ is $29\ \text{km h}^{-1}$ on a bearing of $\alpha$ , where $\tan\alpha = \dfrac{21}{20}$