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

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

Given

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

and

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

, find

\mathbf{a} + \mathbf{b}

and

\mathbf{a} - \mathbf{b}

.

●●●●●Level 22 marksStart

Magnitude and direction of a vector

40 questions

LessonNot started

Relative to a fixed origin

O

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

Find

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

.

●●●●●Level 22 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