Vectors - IB Math AA Question Bank

Question 1

HL • Paper 1

Easy

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HL • Paper 1

Easy

Non Calculator

The points $A$ and $B$ have position vectors

\vec{OA}=\begin{pmatrix}-1\\2\\3\end{pmatrix},\qquad \vec{OB}=\begin{pmatrix}3\\-1\\5\end{pmatrix}

A

Find $\vec{AB}$ and $|\vec{AB}|$ .

[2]

B

Find a unit vector in the direction from $B$ to $A$ .

[2]

C

The point $P$ lies on $AB$ such that $AP:PB=3:1$ . Find the coordinates of $P$ .

[1]

Question 2

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

The line $L$ passes through the points $A(2,-1,3)$ and $B(5,-1,-1)$ .

A

Find a vector equation of $L$ .

[2]

B

Write $L$ in Cartesian form.

[2]

C

Determine whether the point $C(11,-1,-9)$ lies on $L$ .

[1]

Question 3

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

The points $A$ and $B$ have coordinates $A(-2,1,4)$ and $B(4,-3,7)$ . A point $C$ lies on the ray from $A$ through $B$ , and $AC=10$ .

A

Find $\overrightarrow{AB}$ and $|\overrightarrow{AB}|$ .

[2]

B

Find the unit vector in the direction of $\overrightarrow{AB}$ .

[1]

C

Find the coordinates of $C$ .

[2]

Question 4

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

Consider the lines

L_1:\mathbf{r}=\begin{pmatrix}3\\-1\\4\end{pmatrix}+s\begin{pmatrix}2\\-2\\1\end{pmatrix},\quad L_2:\mathbf{r}=\begin{pmatrix}a\\5\\1\end{pmatrix}+t\begin{pmatrix}6\\-6\\3\end{pmatrix}

A

State why $L_1$ and $L_2$ are parallel or coincident.

[1]

B

Find the value of $a$ for which $L_1$ and $L_2$ are coincident.

[3]

C

Determine the relationship between $L_1$ and $L_2$ when $a=9$ .

[1]

Question 5

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

\mathbf{u}=\begin{pmatrix}2\\p\\-1\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}p\\1\\3\end{pmatrix}

A

Given that $\mathbf{u}$ is perpendicular to $\mathbf{v}$ , find $p$ .

[2]

B

For this value of $p$ , let $\theta$ be the angle between $\mathbf{u}$ and $\mathbf{a}=\begin{pmatrix}1\\-2\\2\end{pmatrix}$ . Find the exact value of $\cos\theta$ .

[3]

Question 6

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Two particles $P$ and $Q$ move in three dimensions. At time $t$ seconds, their position vectors are

\mathbf{r}_P=\begin{pmatrix}1\\0\\2\end{pmatrix}+t\begin{pmatrix}2\\-1\\1\end{pmatrix},\qquad \mathbf{r}_Q=\begin{pmatrix}7\\-6\\8\end{pmatrix}+t\begin{pmatrix}-1\\2\\-2\end{pmatrix}

where $t\geq 0$ .

A

Find the speed of each particle.

[2]

B

Determine whether the particles collide. If they do, find the time of collision.

[3]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

\mathbf{u}=\begin{pmatrix}1\\2\\-1\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}3\\-1\\2\end{pmatrix}

A

Find $\mathbf{u}\times\mathbf{v}$ .

[2]

B

Find the area of the parallelogram with adjacent sides represented by $\mathbf{u}$ and $\mathbf{v}$ .

[1]

C

The points $A(1,0,2)$ , $B(2,2,1)$ and $C(4,-1,4)$ are such that $\vec{AB}=\mathbf{u}$ and $\vec{AC}=\mathbf{v}$ . Find the area of triangle $ABC$ .

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The line $L$ and plane $\Pi$ are given by

L:\mathbf{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\2\end{pmatrix},\qquad \Pi:2x-y+2z=3

A

Find the point of intersection of $L$ and $\Pi$ .

[3]

B

Let $\phi$ be the acute angle between $L$ and $\Pi$ . Find the exact value of $\sin\phi$ .

[2]

Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The plane $\Pi$ and line $L$ are given by

\Pi:x+ay+2z=4

L:\mathbf{r}=\begin{pmatrix}2\\0\\1\end{pmatrix}+\lambda\begin{pmatrix}1\\-1\\a\end{pmatrix},\qquad \lambda,a\in\mathbb{R}

A

Find the value of $a$ for which the direction of $L$ is parallel to $\Pi$ .

[2]

B

For this value of $a$ , determine whether $L$ lies in $\Pi$ or is parallel and distinct from $\Pi$ .

[2]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

\mathbf{u}=\begin{pmatrix}2\\-1\\3\end{pmatrix},\quad \mathbf{v}=\begin{pmatrix}k\\4\\1\end{pmatrix}

where $k>0$ . The angle between $\mathbf{u}$ and $\mathbf{v}$ is $1.10$ radians.

A

Write down $\mathbf{u}\cdot \mathbf{v}$ in terms of $k$ .

[1]

B

Find the value of $k$ .

[4]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The line $L_1$ passes through the points $A(1,2,-1)$ and $B(4,-2,5)$ . The line $L_2$ has equation

\mathbf{r}=\begin{pmatrix}2\\1\\0\end{pmatrix}+t\begin{pmatrix}1\\2\\-2\end{pmatrix}

A

Find a vector equation for $L_1$ .

[2]

B

Write $L_1$ in Cartesian form.

[2]

C

Find the acute angle between $L_1$ and $L_2$ .

[2]

Question 12

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The lines $L_1$ and $L_2$ are given by

L_1:\mathbf{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\3\end{pmatrix},\quad L_2:\mathbf{r}=\begin{pmatrix}5\\m\\8\end{pmatrix}+\mu\begin{pmatrix}1\\1\\2\end{pmatrix}

A

Show that, if the lines intersect, then $\lambda=2$ .

[2]

B

Find the value of $m$ for which the lines intersect.

[2]

C

Find the point of intersection when $m=-2$ .

[1]

Question 13

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The points $A$ , $B$ and $C$ have coordinates $A(2,-1,0)$ , $B(5,1,4)$ and $C(0,3,2)$ .

A three-dimensional sketch of triangle ABC with vertices labelled A, B and C. The sides AB and AC should be shown as adjacent vectors from A, with no numerical cross product or area displayed.

A

Find $\rightarrow{AB}\times \rightarrow{AC}$ .

[2]

B

Find the area of triangle $ABC$ .

[2]

C

Find a unit vector perpendicular to the plane containing $A$ , $B$ and $C$ .

[2]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A plane $\i$ has vector equation

\mathbf{r}=\begin{pmatrix}1\\2\\-3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\0\end{pmatrix}+\mu\begin{pmatrix}-1\\3\\2\end{pmatrix}

The point $P(k,1,0)$ lies on $\i$ .

A

Find a normal vector to $\i$ .

[2]

B

Find the Cartesian equation of $\i$ .

[2]

C

Find $k$ .

[1]

Question 15

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The line $L$ and plane $\pi$ are given by

L:\mathbf{r}=\begin{pmatrix}2\\-1\\4\end{pmatrix}+t\begin{pmatrix}1\\3\\-2\end{pmatrix},\quad \pi:2x-y+z=7

A

Find the point where $L$ intersects $\pi$ .

[3]

B

Find the acute angle between $L$ and $\pi$ .

[2]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the lines

L_1:\mathbf{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\3\end{pmatrix},\qquad L_2:\mathbf{r}=\begin{pmatrix}3\\1\\k\end{pmatrix}+\mu\begin{pmatrix}1\\1\\-1\end{pmatrix}

where $\lambda,\mu,k\in\mathbb{R}$ .

A

Show that the two direction vectors are not parallel.

[1]

B

Find the value of $k$ for which $L_1$ and $L_2$ intersect.

[4]

C

State the relationship between $L_1$ and $L_2$ when $k\ne 2$ .

[1]

Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The plane $\Pi$ passes through the points $A(1,0,-1)$ , $B(2,3,0)$ and $C(-1,1,2)$ .

A

Find two non-parallel direction vectors in $\Pi$ .

[2]

B

Find a normal vector to $\Pi$ .

[2]

C

Hence find a Cartesian equation of $\Pi$ .

[2]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The planes $\Pi_1$ and $\Pi_2$ are given by

\Pi_1:x+z=2

\Pi_2:px+y+z=1

where $p\in\mathbb{R}$ . The acute angle between the planes is $\dfrac{\pi}{4}$ .

A

Determine the value of $p$ .

[4]

Question 19

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The line $L$ has equation

L:\mathbf{r}=\begin{pmatrix}1\\-1\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad \lambda\in\mathbb{R}

The point $P$ has coordinates $(0,3,1)$ . A plane $\Pi$ contains $L$ and the point $P$ .

A

Find a vector in $\Pi$ which is not parallel to $L$ .

[1]

B

Find a normal vector to $\Pi$ .

[2]

C

Hence find a Cartesian equation of $\Pi$ .

[2]

Question 20

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The planes $\Pi_1$ and $\Pi_2$ are given by

\Pi_1:x+y+z=3

\Pi_2:2x-y+z=1

The planes intersect in the line $L$ .

A

Find a direction vector for $L$ .

[2]

B

Find one point on $L$ .

[2]

C

Hence write a vector equation of $L$ .

[2]

Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Two drones, $A$ and $B$ , move in straight lines. Their position vectors, in kilometres, $t$ hours after 09:00 are

\mathbf{r}_A=\begin{pmatrix}1\\4\\0\end{pmatrix}+t\begin{pmatrix}2\\-1\\1\end{pmatrix},\quad \mathbf{r}_B=\begin{pmatrix}7\\-2\\3\end{pmatrix}+t\begin{pmatrix}-1\\2\\0\end{pmatrix}

where $0\leq t\leq 4$ .

Drone	Initial position vector / km	Velocity vector / $\text{km h}^{-1}$
A	$(1,4,0)$	$(2,-1,1)$
B	$(7,-2,3)$	$(-1,2,0)$

A

Find $\mathbf{r}_B-\mathbf{r}_A$ in terms of $t$ .

[2]

B

Find the minimum distance between the drones during this time interval.

[3]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The planes $\i_1$ and $\i_2$ are given by

\i_1:x+2y-kz=5,\quad \i_2:3x-y+2z=7

where $k>1$ . The acute angle between the planes is $1.10$ radians.

A

Write down normal vectors to $\i_1$ and $\i_2$ .

[1]

B

Find $k$ .

[4]

Question 23

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The planes $\i_1$ and $\i_2$ are given by

\i_1:x+2y-z=4,\quad \i_2:3x-y+2z=1

The planes intersect in the line $L$ .

A

Find a direction vector for $L$ .

[2]

B

Find a vector equation for $L$ .

[3]

Question 24

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The points $A(1,2,0)$ , $B(3,0,1)$ and $C(0,1,4)$ lie in the plane $\Pi$ . A point $D$ lies on the line through $A$ perpendicular to $\Pi$ .

A

I.

Find $\vec{AB}$ and $\vec{AC}$ .

[2]

II.

Hence find a normal vector to $\Pi$ .

[3]

B

Find a Cartesian equation of $\Pi$ .

[3]

C

Given that $AD=3\sqrt{146}$ and that the $z$ -coordinate of $D$ is positive, find the coordinates of $D$ .

[3]

Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The line $L$ has equation $\mathbf r=\begin{pmatrix}2\\-1\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-2\end{pmatrix}$ , $\lambda\in\mathbb R$ . The point $P$ has coordinates $(5,0,1)$ .

A

I.

Let $A$ be a general point on $L$ . Write down $\vec{PA}$ in terms of $\lambda$ .

[2]

II.

Find the value of $\lambda$ for which $PA$ is perpendicular to $L$ .

[2]

B

Hence find the shortest distance from $P$ to $L$ .

[2]

C

The point $R$ lies on $L$ and $PR=\sqrt{14}$ . Find the possible coordinates of $R$ .

[3]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The lines $L_1$ and $L_2$ are defined by $L_1:\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}$ and $L_2:\mathbf r=\begin{pmatrix}3\\k\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-1\\2\end{pmatrix}$ , where $k\in\mathbb R$ .

A

I.

Show that $L_1$ and $L_2$ are not parallel.

[1]

II.

Find the exact value of the cosine of the acute angle between $L_1$ and $L_2$ .

[2]

B

Find the value of $k$ for which $L_1$ and $L_2$ intersect, and find the point of intersection.

[4]

C

State the relationship between $L_1$ and $L_2$ when $k\ne1$ .

[1]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Two lines are given by $L_1:\mathbf r=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-1\end{pmatrix}$ and $L_2:\mathbf r=\begin{pmatrix}1\\0\\3\end{pmatrix}+\mu\begin{pmatrix}1\\2\\-1\end{pmatrix}$ , where $\lambda,\mu\in\mathbb R$ .

A

I.

State why the lines are parallel.

[1]

II.

Show that the lines are distinct.

[2]

B

Find a Cartesian equation of the plane containing both $L_1$ and $L_2$ .

[3]

C

Find the exact distance between $L_1$ and $L_2$ .

[3]

Question 28

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Two inspection robots, $A$ and $B$ , move along straight paths inside a storage facility. Their position vectors, in metres, $t$ minutes after 10:00, are

\mathbf{r}_A=\begin{pmatrix}2\\1\\0\end{pmatrix}+t\begin{pmatrix}3\\-1\\2\end{pmatrix},\qquad \mathbf{r}_B=\begin{pmatrix}12\\-5\\4\end{pmatrix}+t\begin{pmatrix}-1\\2\\1\end{pmatrix}

where $0\leq t\leq 4$ .

A

I.

Find the speed of each robot.

[2]

II.

Find the acute angle between the two paths.

[2]

B

Let $D(t)$ be the distance between the robots at time $t$ . Show that $D(t)^2=26t^2-124t+152$ .

[3]

C

Hence find the minimum distance between the robots during the four-minute interval. State whether they come within $2$ metres of each other.

[3]

Question 29

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A triangular metal plate has vertices $A(1,2,-1)$ , $B(5,0,2)$ and $C(2,6,1)$ . A laser path is modelled by the infinite line

L:\mathbf{r}=\begin{pmatrix}8\\7\\5\end{pmatrix}+\lambda\begin{pmatrix}-2\\-1\\-3\end{pmatrix}

A 3D schematic showing triangular metal plate $ABC$ lying within a larger translucent plane, with an infinite straight line labelled $L$ intersecting the plane at a point outside the finite triangle, on an extension beyond the plate. Show the line direction with an arrow, but do not show intersection coordinates or suggest that the line enters the triangular plate.

A

I.

Find $\vec{AB}\times\vec{AC}$ .

[2]

II.

Find the area of triangle $ABC$ .

[2]

III.

Find a Cartesian equation of the plane containing the plate.

[1]

B

Find the point where the line $L$ meets the plane of the plate.

[3]

C

Find the acute angle between the line $L$ and the plate.

[2]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The line $L$ is given by

L:\mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}

A plane $\pi$ contains $L$ and the point $P(4,0,5)$ .

A three-dimensional sketch showing a plane containing a labelled line L and a labelled point P not on the line. The direction of L should be indicated, with no normal vector or angle values shown.

A

Find two non-parallel direction vectors in $\pi$ .

[2]

B

Find the Cartesian equation of $\pi$ .

[3]

C

Find the acute angle between $\pi$ and the plane $x+y-2z=0$ .

[2]

Question 31

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A navigation system records three points in space, $A(-6,2,-1)$ , $B(0,1,1)$ and $C(-4,6,-2)$ . A fourth point $D$ is to be chosen so that $ABDC$ is a parallelogram, with $AB$ and $AC$ as adjacent sides.

A three-dimensional coordinate diagram showing points A, B and C with arrows from A to B and from A to C. The point D is not plotted, but a faint parallelogram outline indicates where it will be completed. Axes are labelled x, y and z.

A

AI.

Find $\vec{AB}$ and $\vec{AC}$ .

[2]

AII.

Determine the coordinates of $D$ .

[2]

B

B.

Show that the diagonals of $ABDC$ bisect each other.

[2]

C

A point $P$ moves along the diagonal $AD$ .

CI.

Show that the value of the parameter $t$ for which $P=A+t\vec{AD}$ is closest to the origin is $\frac{43}{74}$ .

[3]

CII.

Hence find the minimum distance from the origin to the diagonal $AD$ . If you did not obtain $t=\frac{43}{74}$ , use this value.

[1]

Question 32

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Two particles $A$ and $B$ move in straight lines. At time $t$ seconds, their position vectors are $\mathbf r_A=\begin{pmatrix}1\\2\\0\end{pmatrix}+t\begin{pmatrix}2\\-1\\1\end{pmatrix}$ and $\mathbf r_B=\begin{pmatrix}5\\1\\0\end{pmatrix}+t\begin{pmatrix}-1\\1\\0\end{pmatrix}$ , where $t\geq0$ .

A

I.

Write down $\vec{AB}$ in terms of $t$ .

[2]

II.

Show that $AB^2=14t^2-28t+17$ .

[2]

B

Find the minimum distance between the particles and the time at which it occurs.

[3]

C

Find the values of $t$ for which $AB=\sqrt7$ .

[3]

Question 33

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The planes $\Pi_1$ and $\Pi_2$ have equations $\Pi_1:x+2y-z=4$ and $\Pi_2:2x-y+z=0$ . Their line of intersection is $L$ .

A

I.

Find a direction vector for $L$ .

[3]

II.

Show that $(1,1,-1)$ lies on $L$ .

[2]

B

Write down a vector equation of $L$ .

[2]

C

Find the exact value of the cosine of the acute angle between $\Pi_1$ and $\Pi_2$ .

[2]

D

Find the Cartesian equation of the plane which contains $L$ and the point $Q(1,0,0)$ .

[3]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The plane $\Pi$ has vector equation $\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+s\begin{pmatrix}1\\1\\-1\end{pmatrix}+t\begin{pmatrix}2\\-1\\1\end{pmatrix}$ , where $s,t\in\mathbb R$ . The line $L$ is given by $\mathbf r=\begin{pmatrix}3\\-2\\5\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\-1\end{pmatrix}$ .

A

I.

Find a normal vector to $\Pi$ .

[3]

II.

Find a Cartesian equation of $\Pi$ .

[2]

B

Find the point of intersection of $L$ and $\Pi$ .

[2]

C

Find the exact value of $\sin\phi$ , where $\phi$ is the acute angle between $L$ and $\Pi$ .

[2]

D

Find the Cartesian equation of the plane which contains $L$ and is perpendicular to $\Pi$ .

[2]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The points $A(1,0,0)$ , $B(3,1,-1)$ , $D(p,1,3)$ and $C$ form a parallelogram $ABCD$ , where $C=B+D-A$ .

A

I.

Find $\vec{AB}$ and $\vec{AD}$ in terms of $p$ .

[2]

II.

Find the value of $p$ for which $ABCD$ is a rectangle.

[2]

B

For $p=2$ , find the area of $ABCD$ .

[3]

C

For $p=2$ , find a Cartesian equation of the plane containing the rectangle.

[3]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The plane $\Pi$ has equation $2x-y+2z=5$ . The point $P$ has coordinates $(4,1,2)$ .

A

I.

Write down a vector equation of the line through $P$ perpendicular to $\Pi$ .

[2]

II.

Find the coordinates of the foot $H$ of the perpendicular from $P$ to $\Pi$ .

[2]

B

Find the distance from $P$ to $\Pi$ .

[2]

C

The point $P'$ is the reflection of $P$ in $\Pi$ . Find the coordinates of $P'$ .

[3]

Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Two intersecting lines $L_1$ and $L_2$ pass through $A(1,2,-1)$ . Their direction vectors are $\mathbf d_1=\begin{pmatrix}1\\-1\\2\end{pmatrix}$ and $\mathbf d_2=\begin{pmatrix}2\\1\\-1\end{pmatrix}$ respectively. The plane containing both lines is $\Pi$ . The plane $\Sigma$ has equation $x+y-z=0$ .

A

I.

Find a normal vector to $\Pi$ .

[2]

II.

Find a Cartesian equation of $\Pi$ .

[2]

B

Find the exact value of the cosine of the acute angle between $\Pi$ and $\Sigma$ .

[2]

C

Find a vector equation of the line of intersection of $\Pi$ and $\Sigma$ .

[4]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two straight service cables are modelled by the lines

L_1:\mathbf{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad L_2:\mathbf{r}=\begin{pmatrix}4\\5\\-1\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\2\end{pmatrix}

A 3D sketch of two non-parallel cables labelled L1 and L2 with a short segment drawn between them at right angles to both cables. No coordinates or parameter values are shown.

A

I.

Show that the two lines do not intersect.

[3]

II.

State the geometrical relationship between the two lines.

[1]

B

Let $P$ be the point on $L_1$ and $Q$ the point on $L_2$ such that $PQ$ is perpendicular to both lines. Find the coordinates of $P$ and $Q$ .

[5]

C

Hence find the shortest distance between the two cables and the acute angle between them.

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The planes $\Pi_1$ and $\Pi_2$ are given by

\Pi_1:x-2y+z=3,\qquad \Pi_2:2x+y-z=4

They intersect in the line $L$ . The point $A$ has coordinates $(4,1,7)$ .

A

I.

Find a direction vector for $L$ .

[2]

II.

Find a vector equation of $L$ .

[3]

B

Find the point on $L$ closest to $A$ . If you did not obtain a vector equation for $L$ , use $\mathbf{r}=\begin{pmatrix}11/5\\-2/5\\0\end{pmatrix}+t\begin{pmatrix}1\\3\\5\end{pmatrix}$ .

[3]

C

Find the acute angle between $\Pi_1$ and $\Pi_2$ .

[2]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A parallelogram-shaped sensor screen lies in the plane

\Pi:\mathbf{r}=\begin{pmatrix}2\\-1\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\0\end{pmatrix}+\mu\begin{pmatrix}0\\1\\2\end{pmatrix}

where $0\leq \lambda\leq 3$ and $0\leq \mu\leq 2$ . A signal travels along

L:\mathbf{r}=\begin{pmatrix}10\\5\\8\end{pmatrix}+t\begin{pmatrix}-3\\-2\\-1\end{pmatrix}

A tilted parallelogram-shaped screen in 3D with two clearly labelled parameter directions, $\begin{pmatrix}1\\2\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\\2\end{pmatrix}$, and a straight incoming signal line. The diagram should show the parallelogram boundaries and vertices, use legible labels without overlap or corrupted fragments, and not show the intersection point.

A

I.

Find a normal vector to $\Pi$ .

[2]

II.

Show that a Cartesian equation of $\Pi$ is $4x-2y+z=13$ .

[2]

B

Find the point where the signal line meets the plane of the screen.

[3]

C

Determine whether the signal hits the parallelogram-shaped screen. Justify your answer.

[3]

Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The points $A(1,2,0)$ , $B(7,5,2)$ , $C(5,11,5)$ and $D(-1,8,3)$ are vertices of a quadrilateral. The point $E$ has coordinates $(2,4,9)$ .

A

I.

Show that $ABCD$ is a parallelogram.

[2]

II.

Find the area of $ABCD$ .

[2]

B

Find a Cartesian equation of the plane containing $ABCD$ .

[3]

C

Find the shortest distance from $E$ to the plane containing $ABCD$ . If you did not obtain the plane equation, use $-3x-22y+42z+47=0$ .

[4]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two underwater probes move with constant velocities. Probe $A$ starts from $(0,1,-2)$ and has actual velocity $\begin{pmatrix}4\\-1\\1\end{pmatrix}$ metres per second. Probe $B$ starts from $(18,-2,5)$ and has actual velocity $\begin{pmatrix}-2\\1\\-1\end{pmatrix}$ metres per second. Let $t$ be the time in seconds after both probes start moving.

A

I.

Write down vector equations for the positions of the two probes.

[2]

II.

Find the acute angle between the paths of the two probes.

[2]

B

Determine whether the probes collide.

[2]

C

Find the time at which the probes are closest together, and find this minimum distance.

[4]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two straight tunnels are modelled by

L_1:\mathbf{r}=\begin{pmatrix}2\\-1\\0\end{pmatrix}+s\begin{pmatrix}1\\2\\1\end{pmatrix}

L_2:\mathbf{r}=\begin{pmatrix}5\\5\\3\end{pmatrix}+u\begin{pmatrix}-2\\1\\0\end{pmatrix}

A

I.

Show that the tunnels intersect, and find their point of intersection.

[3]

II.

Find the angle between the two tunnels.

[1]

B

Find a Cartesian equation of the plane containing both tunnels.

[3]

C

A ventilation shaft is drilled from $A(1,1,8)$ perpendicular to this plane. Find the point where the shaft meets the plane, and the length of the shaft.

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two underwater vehicles move in straight lines. Their position vectors, in kilometres, $t$ hours after noon are

\mathbf{r}_P=\begin{pmatrix}2\\-3\\1\end{pmatrix}+t\begin{pmatrix}1\\2\\-1\end{pmatrix},\qquad \mathbf{r}_Q=\begin{pmatrix}8\\5\\-2\end{pmatrix}+t\begin{pmatrix}-2\\0\\1\end{pmatrix}

where $0\leq t\leq 5$ .

Vehicle	$x(t)$ / km	$y(t)$ / km	$z(t)$ / km	Time interval / h
P	$2+t$	$-3+2t$	$1-t$	$0\le t\le 5$
Q	$8-2t$	$5$	$-2+t$	$0\le t\le 5$

A

I.

Find the speed of each vehicle.

[2]

B

I.

Show that the square of the distance between the vehicles is $17t^2-80t+109$ .

[3]

II.

Hence find the minimum distance between the vehicles during the time interval.

[3]

C

For safety, the vehicles must remain at least $4$ km apart.

I.

Determine the interval of time during which the safety condition (the vehicles' separation is at least $4\ \text{km}$ ) is not satisfied.

[3]

II.

Find the duration of this unsafe interval.

[1]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Consider the two lines

L_1:\mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad L_2:\mathbf{r}=\begin{pmatrix}a\\0\\1\end{pmatrix}+\mu\begin{pmatrix}1\\3\\-2\end{pmatrix}

where $a,\lambda,\mu\in\mathbb{R}$ .

A three-dimensional sketch showing exactly two non-parallel lines, labelled $L_1$ and $L_2$, in a representative skew configuration. Draw only one line $L_2$; do not show duplicate or alternative positions of $L_2$, and do not suggest a third line.

A

I.

Show that $L_1$ and $L_2$ are not parallel.

[1]

II.

Find the value of $a$ for which the two lines intersect, and find the point of intersection.

[5]

B

I.

State the relationship between the lines when $a\ne 5$ .

[1]

C

Assume $a\ne5$ .

I.

Find the shortest distance between $L_1$ and $L_2$ in terms of $a$ .

[4]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The point $C_s=(2,s,3)$ moves along a straight vertical track. The fixed points are $A(1,0,2)$ and $B(4,1,-1)$ . The area of triangle $ABC_s$ changes as $s$ varies.

A corrected three-dimensional coordinate diagram showing exactly three coordinate axes, labelled $x$, $y$, and $z$, with no additional horizontal ray. A vertical track, distinct from the coordinate axes, contains the points $C_s$. Fixed points $A$ and $B$ are shown, along with several faint possible positions of $C_s$. The triangle $ABC_s$ is shaded for one representative value of $s$.

I.

Find $\vec{AB}\times\vec{AC_s}$ in terms of $s$ .

[3]

I.

Show that the area of triangle $ABC_s$ is $\frac{1}{2}\sqrt{18s^2+38}$ .

[2]

II.

Hence find the minimum possible area of triangle $ABC_s$ .

[2]

I.

Find the values of $s$ for which the area of triangle $ABC_s$ is $5$ square units.

[3]

II.

Explain geometrically why two different values of $s$ give the same area in part (c)(i).

[2]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A family of planes $\Pi_k$ passes through the point $A(1,-1,2)$ and contains the two direction vectors

\mathbf{u}=\begin{pmatrix}2\\1\\0\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}0\\k\\3\end{pmatrix}

A three-dimensional sketch of a plane through point A. Two non-parallel direction arrows in the plane are labelled u and v, with the second direction depending on k. A normal vector is shown perpendicular to the plane.

A

AI.

Find a normal vector to $\Pi_k$ in terms of $k$ .

[2]

AII.

Show that a Cartesian equation of $\Pi_k$ is $3x-6y+2kz=9+4k$ .

[2]

B

B.

Find the value of $k$ for which $P(5,0,1)$ lies on $\Pi_k$ .

[2]

C

For the remainder of this question, take $k=3$ .

CI.

Find the acute angle between $\Pi_3$ and the plane $2x+y-z=4$ .

[3]

CII.

Find the point on $\Pi_3$ closest to the origin, and hence find the shortest distance from the origin to $\Pi_3$ .

[3]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two planes are given by

\Pi_1:x-y+2z=4

\Pi_2:2x+y-z=1

They intersect in a line $L$ .

A three-dimensional sketch of two non-parallel planes $\Pi_1$ and $\Pi_2$ intersecting in a single line $L$. Show the normals to both planes and highlight the line of intersection. Label the intersection line $L$ exactly once; do not include a second $L$ label or any leader pointing to a plane boundary.

A

I.

Find a direction vector for $L$ .

[2]

II.

Find a vector equation of $L$ .

[3]

B

For each real value of $k$ , define the plane

\Pi_k:(1+2k)x+(-1+k)y+(2-k)z=4+k

I.

Show that every plane $\Pi_k$ contains the line $L$ .

[2]

II.

Find $k$ such that $\Pi_k$ is perpendicular to the plane $x+z=0$ .

[2]

C

I.

Find the acute angle between $\Pi_1$ and $\Pi_2$ .

[3]

Question 49

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The lines $L_1$ and $L_2$ are given by $L_1:\mathbf r=\begin{pmatrix}0\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\0\\0\end{pmatrix}$ and $L_2:\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\mu\begin{pmatrix}0\\1\\1\end{pmatrix}$ .

A

I.

Show that the two lines do not intersect.

[2]

II.

State the relationship between the two lines.

[1]

B

Let $A$ be a point on $L_1$ and $B$ be a point on $L_2$ such that $AB$ is perpendicular to both lines. Find the coordinates of $A$ and $B$ .

[4]

C

Hence find the shortest distance between $L_1$ and $L_2$ .

[1]

D

Find the Cartesian equation of the plane containing $L_1$ and parallel to $L_2$ .

[2]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Three planes are given by $\Pi_1:x+y+z=3$ , $\Pi_2:x-y+2z=4$ and $\Pi_3:2x+ay+5z=b$ , where $a,b\in\mathbb R$ . The line of intersection of $\Pi_1$ and $\Pi_2$ is denoted by $L$ .

A

I.

Find a direction vector for $L$ .

[2]

II.

Find a vector equation of $L$ .

[3]

B

Find the values of $a$ and $b$ for which $\Pi_3$ contains $L$ .

[4]

C

For $a=0$ and $b=10$ , find the common point of intersection of the three planes.

[3]

Question 51

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The line $L$ and the plane $\Pi_k$ are defined by

L:\mathbf{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+t\begin{pmatrix}2\\-1\\3\end{pmatrix},\qquad \Pi_k:kx+y-2z=5

where $k\in\mathbb{R}$ .

A

I.

Find the value of $k$ for which $L$ is parallel to $\Pi_k$ .

[2]

II.

For this value of $k$ , determine whether $L$ lies in $\Pi_k$ or is parallel and distinct from $\Pi_k$ .

[2]

B

For $k=1$ , find the point of intersection of $L$ and $\Pi_k$ , and find the acute angle between $L$ and $\Pi_k$ .

[4]

C

Find all values of $k$ for which the acute angle between $L$ and $\Pi_k$ is $0.400$ radians.

[4]

Question 52

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Three planes are defined by

\Pi_1:x+y+z=6

\Pi_2:2x-y+3z=10

\Pi_3:x+ay+2z=7

where $a\in\mathbb{R}$ .

A

I.

Find a vector equation of the line of intersection of $\Pi_1$ and $\Pi_2$ .

[3]

II.

Find the coordinates of the common point of the three planes when $a=1$ .

[1]

B

Determine the value of $a$ for which the three planes have no common point. Justify your answer.

[4]

C

For $a=1$ , find the acute angle between $\Pi_2$ and $\Pi_3$ .

[2]

Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A triangular solar panel has vertices $A(0,0,2)$ , $B(4,1,3)$ and $C(1,5,6)$ . A vertical support cable passes through the point $D(2,2,0)$ and is parallel to the $z$ -axis.

A sloping triangular panel with vertices A, B and C in 3D, and a vertical cable rising from a floor point D toward the panel. The diagram should indicate the horizontal floor but omit the cable endpoint and any intersection point, so it does not reveal whether the cable meets the triangular region.

A

I.

Find a normal vector to the plane of the panel.

[2]

II.

Find the area of the triangular panel.

[2]

III.

Find a Cartesian equation of the plane of the panel.

[1]

B

Find the height at which the vertical cable meets the plane of the panel.

[2]

C

Determine whether the cable meets the triangular panel itself, rather than only the plane containing it.

[2]

D

Find the acute angle between the panel and the horizontal plane $z=0$ .

[2]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A tunnel is modelled by the line

L_m:\mathbf{r}=\begin{pmatrix}m\\1\\0\end{pmatrix}+t\begin{pmatrix}1\\m\\2\end{pmatrix}

where $m\in\mathbb{R}$ . A fault plane is modelled by

\Pi:2x-y+z=4

A

I.

Find the value of $m$ for which $L_m$ is parallel to $\Pi$ .

[2]

II.

For this value of $m$ , determine whether $L_m$ lies in $\Pi$ or is parallel and distinct from $\Pi$ .

[2]

B

Find the value of $m$ for which $L_m$ intersects $\Pi$ when $t=1$ . Hence find the point of intersection for this value of $m$ .

[3]

C

For $m=1$ , find the acute angle between $L_m$ and $\Pi$ .

[2]

D

Find all values of $m$ for which the acute angle between $L_m$ and $\Pi$ is $20^\circ$ .

[3]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two signal directions in a three-dimensional tracking device are represented by

\mathbf{u}=\begin{pmatrix}3\\-1\\2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}1\\4\\-2\end{pmatrix}

A third adjustable direction is given by $\mathbf{d}=\begin{pmatrix}1\\k\\2\end{pmatrix}$ , where $k\in\mathbb{R}$ .

A schematic three-dimensional axes diagram with three arrows from the origin labelled u, v and d. No curved adjustment marker or component-direction cue is shown; the adjustable component is the second (y) component, as specified in the stem. No numerical angle values are shown.

A

I.

Find the angle between $\mathbf{u}$ and $\mathbf{v}$ .

[3]

II.

Find the vector projection of $\mathbf{v}$ onto $\mathbf{u}$ .

[2]

B

I.

Find $k$ if $\mathbf{d}$ is perpendicular to $\mathbf{u}+\mathbf{v}$ .

[2]

C

The value of $k$ is now allowed to vary.

I.

Determine the value of $k$ for which the acute angle between $\mathbf{d}$ and $\mathbf{u}$ is a minimum.

[4]

II.

Find this minimum acute angle. If you did not obtain $k=-\frac{5}{7}$ , use this value.

[1]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A light ray travels along the line

L:\mathbf{r}=\begin{pmatrix}5\\0\\1\end{pmatrix}+t\begin{pmatrix}-2\\1\\2\end{pmatrix},\qquad t\in\mathbb{R}

and reflects from the plane mirror

\Pi:x+2y-2z=6

A three-dimensional diagram showing a plane mirror labelled Pi and an incoming ray labelled L meeting the plane. A normal vector to the plane is drawn at the point of contact, and a reflected ray is indicated without numerical labels.

A

I.

Find the point at which the ray meets the plane.

[3]

II.

Find the acute angle between the incoming ray and the plane.

[3]

B

The reflected direction is obtained by reversing the component of the incoming direction parallel to the normal vector of the mirror.

I.

Find a direction vector for the reflected ray.

[4]

II.

Write down a vector equation of the reflected ray.

[2]

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two straight support cables are modelled by the following two lines

L_1:\mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\1\end{pmatrix}

L_2:\mathbf{r}=\begin{pmatrix}3\\-1\\0\end{pmatrix}+\mu\begin{pmatrix}2\\1\\-1\end{pmatrix}

A three-dimensional diagram showing two straight lines labelled $L_1$ and $L_2$. A shortest connecting segment $PQ$ is drawn perpendicular to both lines.

A

I.

Show that the two lines are skew.

[3]

II.

Find a unit vector perpendicular to both lines.

[2]

B

I.

Find the shortest distance between the two lines.

[3]

C

Let $P$ be the point on $L_1$ and $Q$ the point on $L_2$ such that $PQ$ is the shortest connecting segment.

I.

Find the coordinates of $P$ and $Q$ .

[5]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A tetrahedral frame has vertices $A(1,0,0)$ , $B(3,2,1)$ , $C(0,4,2)$ and $D(2,-1,h)$ , where $h\in\mathbb{R}$ and $h\ne-\frac{1}{2}$ .

A three-dimensional diagram of tetrahedron ABCD with coordinate axes. The vertex $D(2,-1,h)$ is shown moving parallel to the $z$-axis. Do not draw a height or perpendicular arrow from D to face $ABC$; $h$ labels the $z$-coordinate only. Face $ABC$ is shaded.

A

A.

Find $\vec{AC}\times\vec{AD}$ in terms of $h$ .

[3]

B.

Show that the volume $V$ of the tetrahedron is $\frac{|10h+5|}{6}$ .

[2]

B

C.

Find the values of $h$ for which the volume is $5$ cubic units.

[3]

C

D.

Determine the value of $h$ for which the faces $ABC$ and $ABD$ are perpendicular.

[4]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A triangular landing panel has vertices $A(0,0,0)$ , $B(6,0,2)$ and $C(2,5,1)$ . A probe travels along the line

\mathbf{r}=\begin{pmatrix}1\\2\\8\end{pmatrix}+t\begin{pmatrix}1\\-1\\-3\end{pmatrix},\qquad t\geq 0

A three-dimensional diagram showing the triangular panel ABC in its plane and a straight probe path. The intersection point $P=\left(\frac{162}{49},-\frac{15}{49},\frac{53}{49}\right)$ is shown on the plane but outside the triangular panel, on the side of $AB$ opposite $C$, with the edges $AB$, $BC$ and $CA$ labelled.

A

I.

Find a Cartesian equation of the plane containing the triangular panel.

[3]

B

I.

Find the point where the probe path meets the plane.

[3]

II.

Determine whether this point lies inside the triangular panel.

[3]

C

A second probe travels in the same direction as the first probe, starting from $(1,q,8)$ , where $q$ is a real constant. Its path is

\mathbf{r}=\begin{pmatrix}1\\q\\8\end{pmatrix}+t\begin{pmatrix}1\\-1\\-3\end{pmatrix},\qquad t\geq 0

I.

Find $q$ so that the second probe hits the side $AB$ of the triangular panel.

[4]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two planes are given by

\Pi_1:2x-y+2z=3

\Pi_2:x+2y-2z=-1

A moving point lies on the line

L:\mathbf{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+s\begin{pmatrix}2\\-1\\1\end{pmatrix},\qquad s\in\mathbb{R}

A three-dimensional diagram showing two intersecting planes, labelled exactly once as $\Pi_1:2x-y+2z=3$ and $\Pi_2:x+2y-2z=-1$, and a line $L$ crossing the region between them. Each leader line points to the corresponding plane, with no duplicate $\Pi_2$ annotation. Two angle-bisector planes are indicated faintly as surfaces on which points are equidistant from $\Pi_1$ and $\Pi_2$.

A

A.

Show that the distances from a point on $L$ to $\Pi_1$ and $\Pi_2$ are $\frac{|3+7s|}{3}$ and $\frac{|2+2s|}{3}$ respectively.

[3]

B

B.

Find the points on $L$ which are equidistant from $\Pi_1$ and $\Pi_2$ .

[3]

C

CI.

Find Cartesian equations of the two planes whose points are equidistant from $\Pi_1$ and $\Pi_2$ .

[3]

CII.

Find the acute angle between $\Pi_1$ and $\Pi_2$ .