Vectors - IB Math AA Question Bank

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Vectors

Practice exam-style IB Math AA questions for Vectors, aligned with the syllabus and grouped by topic.

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Question 1

HL • Paper 1

Easy

Non Calculator

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}

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

[2]

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

[2]

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 $B(5,-1,-1)$ .

Find a vector equation of $L$ .

[2]

Write $L$ in Cartesian form.

[2]

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 $AC=10$ .

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

[2]

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

[1]

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}

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

[1]

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

[3]

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

[1]

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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}

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

[2]

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 $t\geq 0$ .

Find the speed of each particle.

[2]

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}

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

[2]

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

[1]

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

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

[3]

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

[2]

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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}

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

[2]

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.

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

[1]

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}

Find a vector equation for $L_1$ .

[2]

Write $L_1$ in Cartesian form.

[2]

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}

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

[2]

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

[2]

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

[1]

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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 $C(0,3,2)$ .

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

[2]

Find the area of triangle $ABC$ .

[2]

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 $\i$ .

Find a normal vector to $\i$ .

[2]

Find the Cartesian equation of $\i$ .

[2]

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

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

[3]

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

Show that the two direction vectors are not parallel.

[1]

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

[4]

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

[1]

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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 $C(-1,1,2)$ .

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

[2]

Find a normal vector to $\Pi$ .

[2]

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 $\dfrac{\pi}{4}$ .

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 $P$ .

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

[1]

Find a normal vector to $\Pi$ .

[2]

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 $L$ .

Find a direction vector for $L$ .

[2]

Find one point on $L$ .

[2]

Hence write a vector equation of $L$ .

[2]

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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	$(7,-2,3)$	$(-1,2,0)$

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

[2]

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.

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

[1]

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 $L$ .

Find a direction vector for $L$ .

[2]

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 $\Pi$ .

I.

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

[2]

II.

Hence find a normal vector to $\Pi$ .

[3]

Find a Cartesian equation of $\Pi$ .

[3]

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

[3]

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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 $(5,0,1)$ .

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]

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

[2]

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 $k\in\mathbb R$ .

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]

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

[4]

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 $\lambda,\mu\in\mathbb R$ .

I.

State why the lines are parallel.

[1]

II.

Show that the lines are distinct.

[2]

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

[3]

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 $0\leq t\leq 4$ .

I.

Find the speed of each robot.

[2]

II.

Find the acute angle between the two paths.

[2]

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

[3]

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

[3]

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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}

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]

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

[3]

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 $P(4,0,5)$ .

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

[2]

Find the Cartesian equation of $\pi$ .

[3]

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.

AI.

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

[2]

AII.

Determine the coordinates of $D$ .

[2]

B.

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

[2]

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 $t\geq0$ .

I.

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

[2]

II.

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

[2]

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

[3]

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

[3]

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Question 33

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

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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 $L$ .

I.

Find a direction vector for $L$ .

[3]

II.

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

[2]

Write down a vector equation of $L$ .

[2]

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

[2]

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 $\mathbf r=\begin{pmatrix}3\\-2\\5\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\-1\end{pmatrix}$ .

I.

Find a normal vector to $\Pi$ .

[3]

II.

Find a Cartesian equation of $\Pi$ .

[2]

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

[2]

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

[2]

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 $C=B+D-A$ .

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]

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

[3]

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 $(4,1,2)$ .

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]

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

[2]

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

[3]

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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 $x+y-z=0$ .

I.

Find a normal vector to $\Pi$ .

[2]

II.

Find a Cartesian equation of $\Pi$ .

[2]

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

[2]

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}

I.

Show that the two lines do not intersect.

[3]

II.

State the geometrical relationship between the two lines.

[1]

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]

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 $(4,1,7)$ .

I.

Find a direction vector for $L$ .

[2]

II.

Find a vector equation of $L$ .

[3]

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]

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}

I.

Find a normal vector to $\Pi$ .

[2]

II.

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

[2]

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

[3]

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

[3]

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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 $(2,4,9)$ .

I.

Show that $ABCD$ is a parallelogram.

[2]

II.

Find the area of $ABCD$ .

[2]

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

[3]

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.

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]

Determine whether the probes collide.

[2]

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}

I.

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

[3]

II.

Find the angle between the two tunnels.

[1]

Find a Cartesian equation of the plane containing both tunnels.

[3]

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	$8-2t$	$5$	$-2+t$	$0\le t\le 5$

I.

Find the speed of each vehicle.

[2]

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]

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]

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

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]

I.

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

[1]

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.

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}

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.

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

[2]

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 $L$ .

I.

Find a direction vector for $L$ .

[2]

II.

Find a vector equation of $L$ .

[3]

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]

I.

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

[3]

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

I.

Show that the two lines do not intersect.

[2]

II.

State the relationship between the two lines.

[1]

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]

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

[1]

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 $L$ .

I.

Find a direction vector for $L$ .

[2]

II.

Find a vector equation of $L$ .

[3]

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

[4]

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 $k\in\mathbb{R}$ .

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]

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

[4]

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 $a\in\mathbb{R}$ .

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]

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

[4]

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

[2]

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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.

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]

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

[2]

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

[2]

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

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]

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]

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

[2]

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 $k\in\mathbb{R}$ .

I.

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

[3]

II.

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

[2]

I.

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

[2]

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

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]

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]

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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}

I.

Show that the two lines are skew.

[3]

II.

Find a unit vector perpendicular to both lines.

[2]

I.

Find the shortest distance between the two lines.

[3]

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.

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]

C.

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

[3]

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

I.

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

[3]

I.

Find the point where the probe path meets the plane.

[3]

II.

Determine whether this point lies inside the triangular panel.

[3]

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.

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.

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

[3]

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 $\Pi_2$ .

[3]

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