CAIE A-Level Mathematics 3.7 Vectors Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise 3D vector questions involving lines, positions, intersections, lengths, scalar products and geometric conditions.

Exam points

form vector equations of lines from points and direction vectors, then solve for intersections
use vector lengths and scalar products to prove geometric facts or find exact angles
apply component equations to distinguish parallel, intersecting and skew lines

Covered syllabus points

3.7.1—Standard notations for vectors 3.7.2—Vector operations 3.7.3—Position vectors 3.7.4—Straight lines 3.7.5—Whether two lines are parallel 3.7.6—Scalar product

[Maximum number: 9]

The line $l_{1}$ passes through the point (3,1,-6) and is parallel to the vector 2 i+j+4 k.
The line $l_{2}$ passes through the point (-1,3,-6) and is perpendicular to the vector 3 i-2 j+k. The direction vector for $l_{2}$ has no component in the x-direction.

(a)

Write down a vector equation for $l_{1}$ and find a vector equation for $l_{2}$ .

[ 3 ]

(b)

Calculate the acute angle between $l_{1}$ and $l_{2}$ .

[ 3 ]

(c)

Find the position vector of the point of intersection of $l_{1}$ and $l_{2}$ .

[ 3 ]

[Maximum number: 10]

The equations of two lines are given by

\begin{aligned} & l_{1}: \mathbf{r}=(2 \mathbf{i}+\mathbf{j}+4 \mathbf{k})+\lambda(\mathbf{i}+2 \mathbf{j}-3 \mathbf{k}), \\ & l_{2}: \mathbf{r}=(3 \mathbf{i}-\mathbf{j}+5 \mathbf{k})+\mu(2 \mathbf{i}+3 \mathbf{j}+a \mathbf{k}) . \end{aligned}

(a)

Find the value of a for which $l_{1}$ is perpendicular to $l_{2}$ .

[ 2 ]

(b)

Find the value of a for which $l_{1}$ and $l_{2}$ intersect.

[ 4 ]

(c)

Find the values of a for which the acute angle between $l_{1}$ and $l_{2}$ is equal to $\cos ^{-1}\left(\frac{5}{14}\right)$ .

[ 4 ]

[Maximum number: 10]

With respect to the origin O, the points A, B, C and D have position vectors given by

\overrightarrow{O A}=\left(\begin{array}{l} 1 \\ 5 \\ 3 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 0 \\ 4 \\ 1 \end{array}\right), \quad \overrightarrow{O C}=\left(\begin{array}{r} 1 \\ -3 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O D}=\left(\begin{array}{r} 3 \\ -5 \\ 4 \end{array}\right) .

The line m passes through the points A and B.

(a)

Find a vector equation for m.

[ 2 ]

(b)

Find the position vector of the point of intersection of m and the line passing through the points C and D.

[ 4 ]

(c)

Find the position vector of the foot of the perpendicular from C to m.

[ 4 ]

[Maximum number: 8]

With respect to the origin O, the points A, B and C have position vectors given by

\overrightarrow{O A}=2 \mathbf{i}-\mathbf{j}-6 \mathbf{k}, \quad \overrightarrow{O B}=b \mathbf{i}-2 \mathbf{j}+3 \mathbf{k} \quad \text { and } \quad \overrightarrow{O C}=-4 \mathbf{i}+5 \mathbf{j}-2 \mathbf{k} .

(a)

It is given that $|\overrightarrow{A B}|=|\overrightarrow{B C}|$ .

Find the value of b.

[ 3 ]

(b)

A, B, C and D are the vertices of a rhombus.

Find the position vector of D.

[ 2 ]

(c)

Calculate angle ABC.

[ 3 ]

[Maximum number: 5]

Figure for Question 11 — CAIE A-Level Mathematics

Relative to the origin O, the position vectors of the points A, B and C are

\overrightarrow{O A}=4 \mathbf{i}-2 \mathbf{j}, \overrightarrow{O B}=2 \mathbf{i}+8 \mathbf{j}+4 \mathbf{k} \text {, and } \overrightarrow{O C}=-2 \mathbf{i}+6 \mathbf{k}

The midpoint of AB is M, as shown in the diagram.

(a)

Find the vectors $\overrightarrow{M B}$ and $\overrightarrow{M C}$ .

[ 2 ]

(b)

Calculate the exact value of the cosine of angle CMB.

[ 3 ]

Exam points

Covered syllabus points

Question 9

Question 9(a)

Question 9(b)

Question 9(c)

Question 9

Question 9(a)

Question 9(b)

Question 9(c)

Question 11

Question 11(a)

Question 11(b)

Question 11(c)

Question 10

Question 10(a)

Question 10(b)

Question 10(c)

Question 11

Question 11(a)

Question 11(b)