Vectors — Higher Maths Practice Questions

1

Three vectors are defined as follows:

\displaystyle \overrightarrow{\textsf{RS}} = -3\boldsymbol{i}+2\boldsymbol{j}+\boldsymbol{k}

,

\displaystyle \overrightarrow{\textsf{ST}} = \boldsymbol{i}-3\boldsymbol{j}+5\boldsymbol{k}

,

\displaystyle \overrightarrow{\textsf{PT}} = 2\boldsymbol{i}+\boldsymbol{j}-3\boldsymbol{k}

.
(a) Find

\displaystyle \overrightarrow{\textsf{RT}}

.
(b) Hence, or otherwise, find

\displaystyle \overrightarrow{\textsf{RP}}

.

Video solution coming soon

2

PQRS is a trapezium with

\displaystyle \overrightarrow{\textsf{RQ}}=2\,\overrightarrow{\textsf{SP}}

. Let

\displaystyle \overrightarrow{\textsf{PQ}}=\boldsymbol{u}

and

\displaystyle \overrightarrow{\textsf{RQ}}=\boldsymbol{v}

.
(a) Express

\displaystyle \overrightarrow{\textsf{RP}}

in terms of

\displaystyle \boldsymbol{u}

and

\displaystyle \boldsymbol{v}

.
(b) Express

\displaystyle \overrightarrow{\textsf{RS}}

in terms of

\displaystyle \boldsymbol{u}

and

\displaystyle \boldsymbol{v}

, in its simplest form.

Video solution coming soon

3

Show that the points

\displaystyle A(-1,\,3,\,0)

,

\displaystyle B(2,\,-1,\,4)

and

\displaystyle C(-7,\,11,\,-8)

are collinear.

Video solution coming soon

4

Show that the points

\displaystyle P(2,\,-6,\,8)

,

\displaystyle Q(0,\,-5,\,5)

and

\displaystyle R(8,\,-9,\,0)

are \textit{not} collinear.

Video solution coming soon

5

(a) Show that the points

\displaystyle F(-3,\,-5,\,9)

,

\displaystyle G(0,\,1,\,0)

and

\displaystyle H(2,\,5,\,-6)

are collinear.
(b) State the ratio in which G divides FH.

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6

\displaystyle A(9,\,-3,\,-8)

,

\displaystyle B(1,\,t,\,4)

and

\displaystyle C(-1,\,2,\,7)

are collinear.
(a) State the ratio in which B divides AC.
(b) Find the value of

\displaystyle t

.

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7

R and T are the points

\displaystyle (7,-1,8)

and

\displaystyle (-3,4,-7)

respectively. S divides RT internally in the ratio

\displaystyle 2:3

.
Determine the coordinates of point S.

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8

A and B are the points

\displaystyle (-4,1,-3)

and

\displaystyle (0,-6,1)

respectively.

\displaystyle k\,\overrightarrow{\textsf{AB}}

is a unit vector, where

\displaystyle k > 0

.
Determine the value of

\displaystyle k

.

Video solution coming soon

9

Vectors

\displaystyle \boldsymbol{u}=-3\boldsymbol{i}+2\boldsymbol{j}+n\boldsymbol{k}

and

\displaystyle \boldsymbol{v}=2\boldsymbol{i}+5\boldsymbol{j}+2\boldsymbol{k}

are perpendicular.
Determine the value of

\displaystyle n

.

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10

Points A, B and C are

\displaystyle (-7,\,-3,\,-6)

,

\displaystyle (5,\,-2,\,6)

and

\displaystyle (7,\,3,\,-8)

respectively.
Find the angle

\displaystyle \angle\textsf{ABC}

.

Video solution coming soon

11

2015 P1 Q1

2 Marks

Vectors

\displaystyle \mathbf{u}=8\mathbf{i}+2\mathbf{j}-\mathbf{k}

and

\displaystyle \mathbf{v}=-3\mathbf{i}+t\mathbf{j}-6\mathbf{k}

are perpendicular.
Determine the value of

\displaystyle t.

12

2018 P2 Q2

(1, 4)5 Marks

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are defined by
$\displaystyle \mathbf{u}=\begin{pmatrix}-1\\4\\-3\end{pmatrix}$ and $\displaystyle \mathbf{v}=\begin{pmatrix}-7\\8\\5\end{pmatrix}$
(a) Find $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
(b) Calculate the acute angle between $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$

13

2019 P1 Q9

(1, 3, 2)6 Marks

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ have components
$\displaystyle \begin{pmatrix}p\\-2\\4\end{pmatrix}$ and $\displaystyle \begin{pmatrix}2p+16\\-3\\6\end{pmatrix}$
$\displaystyle p\in\mathbb{R}.$

(a)  (i) Find an expression for $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
      (ii) Determine the values of $\displaystyle p$ for which $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are perpendicular.
(b)  Determine the value of $\displaystyle p$ for which $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are parallel.

14

2019 P2 Q14

4 Marks

The vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are such that $\displaystyle |\mathbf{u}|=4$ , $\displaystyle |\mathbf{v}|=5$ and $\displaystyle \mathbf{u}\cdot(\mathbf{u}+\mathbf{v})=21.$
Determine the size of the angle between the vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$

15

2021 P1 Q12

Points A, B and C are collinear, with B dividing AC. A has coordinates

\displaystyle (4,2,-5)

, B has coordinates

\displaystyle (7,-4,1)

,

\displaystyle |\overrightarrow{\textsf{BC}}| = 6

.
(a) (i) Find

\displaystyle |\overrightarrow{\textsf{AB}}|

. (ii) State the ratio in which B divides AC.
(b) Determine the coordinates of C.

Video solution coming soon

16

2021 P2 Q11

(a) Given

\displaystyle A(3,\,1,\,8)

,

\displaystyle B(-2,\,5,\,1)

and

\displaystyle C(7,\,-6,\,3)

, express

\displaystyle \overrightarrow{\textsf{AB}}

and

\displaystyle \overrightarrow{\textsf{AC}}

in component form.
(b) Hence calculate the size of angle BAC.

Video solution coming soon

17

2024 P1 Q4

2 Marks

P and Q have coordinates (-6, 1, 2) and (-1, 11, -8) respectively.
Find the coordinates of the point R which divides PQ in the ratio 2:3.

18

2024 P2 Q3

(2, 1, 4)7 Marks

The coordinates of points D, E and F are given by $\displaystyle D(2,-3,4)$ , $\displaystyle E(1,1,-2)$ and $\displaystyle F(3,2,1).$
(a)  Express $\displaystyle \overrightarrow{ED}$ and $\displaystyle \overrightarrow{EF}$ in component form.
(b)  (i) Calculate $\displaystyle \overrightarrow{ED}.\overrightarrow{EF}.$
      (ii) Hence, or otherwise, calculate the size of angle DEF.

19

2025 P1 Q10

5 Marks

The vectors u and v are such that:
$\displaystyle u=\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$
$\displaystyle v=\begin{pmatrix}1\\ 3\\ k\end{pmatrix}$
the angle between u and v is $\displaystyle 45^{\circ}$ .
Find the value of k, where $\displaystyle k \gt 0$ .

20

2025 P2 Q5

(3, 1)4 Marks

(a) Show that the points $\displaystyle A(-3,2,-1),$ $\displaystyle B(6,-1,5)$ and $\displaystyle C(12,-3,9)$ are collinear.
(b) State the ratio in which B divides AC.

21

2025 P2 Q8

2 Marks

E,ABCD is a rectangular-based pyramid as shown.
$\displaystyle \vec{AD}=6i+4j+2k$
$\displaystyle \vec{DC}=2i-4j+2k$
$\displaystyle \vec{DE}=-4i-3j+4k$

Express $\displaystyle \vec{BE}$ in terms of i, j and k.

22

2026 P1 Q9

(3, 2)5 Marks

(a) Show that the points $\displaystyle D(-1,6,1)$ , $\displaystyle E(1,2,7)$ and $\displaystyle F(2,0,10)$ are collinear.
Point G is such that F divides DG in the ratio 3:4.
(b) Find the coordinates of G.

23

2026 P2 Q2

(1, 4)5 Marks

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are defined as
$\displaystyle \mathbf{u}=\begin{pmatrix}3\\ 2\\ 6\end{pmatrix}$ and $\displaystyle \mathbf{v}=\begin{pmatrix}-5\\ -2\\ 4\end{pmatrix}.$
(a) Find $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
(b) Calculate the acute angle between $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$