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AHL 3.10—Vector definitions - IB Questionbank | RevisionDojo

Practice AHL 3.10—Vector definitions with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Given vectors $\vec{a}=\begin{pmatrix}4 \\ -2\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}1 \\ 3\end{pmatrix}$ , find the vector $\vec{r}$ such that $\vec{r}$ is perpendicular to both $\vec{a}$ and $\vec{b}-\vec{a}$ .

[5]

Question 2

HLPaper 1

Two vectors are given as $\vec{u} = (3, -2, 1)$ and $\vec{v} = (-1, 4, 5)$ .

Find the sum of the two vectors $\vec{u} + \vec{v}$ .

[2]

Calculate the dot product $\vec{u} \cdot \vec{v}$ .

[3]

Find the angle between the vectors using your result from part 2.

[4]

Calculate the vector $2\vec{u}$ .

[2]

Question 3

HLPaper 1

Find the unit vector in the direction of $\vec{b}=\left[\begin{array}{l}6 \\ 8\end{array}\right]$ .

[2]

Question 4

HLPaper 1

Let $\mathbf{u}=\begin{pmatrix}2 \\ -1\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}-3 \\ 4\end{pmatrix}$ .

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

[2]

Question 5

SLPaper 1

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

The line through A and B is perpendicular to the vector $2\mathbf{i} + n\mathbf{k}$ . Find the value of $n$ .

[3]

Find a vector equation of the line that passes through A and B.

[4]

Question 6

HLPaper 1

A triangle has vertices $A(1,2), B(4,6)$ , and $C(k, 10)$ .

Given that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are perpendicular, find the value of $k$ .

[5]

Question 7

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow. The path of the ball is modelled by the equation $\binom{x}{y} = \binom{5}{0} + t \binom{u_x}{u_y - 5t}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$

Find the initial speed of the ball.

[2]

Find the angle of elevation of the ball as it is launched.

[2]

Find the maximum height reached by the ball.

[3]

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

An archer releases an arrow from the point (0, 2). The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed 60 m s⁻¹ and an angle of elevation of 10°. Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 8

HLPaper 1

A submarine is located in a sea at coordinates $(0.8, 1.3, -0.3)$ relative to a ship positioned at the origin $O$. The $x$ direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards.

All distances are measured in kilometres.

The submarine travels with direction vector $\begin{pmatrix} -2 \\ -3 \\ 1 \end{pmatrix}$.

The submarine reaches the surface of the sea at the point $P$.

Assuming the submarine travels in a straight line, write down an equation for the line along which it travels.

[2]

Find the coordinates of $P$.

[3]

Find $OP$.

[2]

Question 9

HLPaper 1

The position vectors of points $A, B$ , and $C$ relative to the origin $O$ are:

\overrightarrow{OA}=\begin{pmatrix} 1 \\ 3 \end{pmatrix}, \quad \overrightarrow{OB}=\begin{pmatrix} 4 \\ 5 \end{pmatrix}, \quad \overrightarrow{OC}=\begin{pmatrix} 7 \\ k \end{pmatrix}

Given that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel, find the value of $k$ .

[4]

Question 10

HLPaper 1

Let $\vec{a}=\begin{pmatrix}2 \\ -1\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}4 \\ -2\end{pmatrix}$ .

Show that $\vec{a}$ and $\vec{b}$ are parallel.

[2]

Paper

Level

Question 1

HLPaper 1

[5]

Question 2

HLPaper 1

Two vectors are given as $\vec{u} = (3, -2, 1)$ and $\vec{v} = (-1, 4, 5)$ .

Find the sum of the two vectors $\vec{u} + \vec{v}$ .

[2]

Calculate the dot product $\vec{u} \cdot \vec{v}$ .

[3]

Find the angle between the vectors using your result from part 2.

[4]

Calculate the vector $2\vec{u}$ .

[2]

Question 3

HLPaper 1

Find the unit vector in the direction of $\vec{b}=\left[\begin{array}{l}6 \\ 8\end{array}\right]$ .

[2]

Question 4

HLPaper 1

Let $\mathbf{u}=\begin{pmatrix}2 \\ -1\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}-3 \\ 4\end{pmatrix}$ .

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

[2]

Question 5

SLPaper 1

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

The line through A and B is perpendicular to the vector $2\mathbf{i} + n\mathbf{k}$ . Find the value of $n$ .

[3]

Find a vector equation of the line that passes through A and B.

[4]

Question 6

HLPaper 1

A triangle has vertices $A(1,2), B(4,6)$ , and $C(k, 10)$ .

Given that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are perpendicular, find the value of $k$ .

[5]

Question 7

HLPaper 2

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$

Find the initial speed of the ball.

[2]

Find the angle of elevation of the ball as it is launched.

[2]

Find the maximum height reached by the ball.

[3]

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

[4]

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 8

HLPaper 1

All distances are measured in kilometres.

The submarine travels with direction vector $\begin{pmatrix} -2 \\ -3 \\ 1 \end{pmatrix}$.

The submarine reaches the surface of the sea at the point $P$.

Assuming the submarine travels in a straight line, write down an equation for the line along which it travels.

[2]

Find the coordinates of $P$.

[3]

Find $OP$.

[2]

Question 9

HLPaper 1

The position vectors of points $A, B$ , and $C$ relative to the origin $O$ are:

\overrightarrow{OA}=\begin{pmatrix} 1 \\ 3 \end{pmatrix}, \quad \overrightarrow{OB}=\begin{pmatrix} 4 \\ 5 \end{pmatrix}, \quad \overrightarrow{OC}=\begin{pmatrix} 7 \\ k \end{pmatrix}

Given that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel, find the value of $k$ .

[4]

Question 10

HLPaper 1

Let $\vec{a}=\begin{pmatrix}2 \\ -1\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}4 \\ -2\end{pmatrix}$ .

Show that $\vec{a}$ and $\vec{b}$ are parallel.

[2]

Paper

Level

Question 1

HLPaper 1

[5]

Question 2

HLPaper 1

Two vectors are given as $\vec{u} = (3, -2, 1)$ and $\vec{v} = (-1, 4, 5)$ .

Find the sum of the two vectors $\vec{u} + \vec{v}$ .

[2]

Calculate the dot product $\vec{u} \cdot \vec{v}$ .

[3]

Find the angle between the vectors using your result from part 2.

[4]

Calculate the vector $2\vec{u}$ .

[2]

Question 3

HLPaper 1

Find the unit vector in the direction of $\vec{b}=\left[\begin{array}{l}6 \\ 8\end{array}\right]$ .

[2]

Question 4

HLPaper 1

Let $\mathbf{u}=\begin{pmatrix}2 \\ -1\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}-3 \\ 4\end{pmatrix}$ .

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

[2]

Question 5

SLPaper 1

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

The line through A and B is perpendicular to the vector $2\mathbf{i} + n\mathbf{k}$ . Find the value of $n$ .

[3]

Find a vector equation of the line that passes through A and B.

[4]

Question 6

HLPaper 1

A triangle has vertices $A(1,2), B(4,6)$ , and $C(k, 10)$ .

Given that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are perpendicular, find the value of $k$ .

[5]

Question 7

HLPaper 2

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$

Find the initial speed of the ball.

[2]

Find the angle of elevation of the ball as it is launched.

[2]

Find the maximum height reached by the ball.

[3]

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

[4]

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 8

HLPaper 1

All distances are measured in kilometres.

The submarine travels with direction vector $\begin{pmatrix} -2 \\ -3 \\ 1 \end{pmatrix}$.

The submarine reaches the surface of the sea at the point $P$.

Assuming the submarine travels in a straight line, write down an equation for the line along which it travels.

[2]

Find the coordinates of $P$.

[3]

Find $OP$.

[2]

Question 9

HLPaper 1

The position vectors of points $A, B$ , and $C$ relative to the origin $O$ are:

\overrightarrow{OA}=\begin{pmatrix} 1 \\ 3 \end{pmatrix}, \quad \overrightarrow{OB}=\begin{pmatrix} 4 \\ 5 \end{pmatrix}, \quad \overrightarrow{OC}=\begin{pmatrix} 7 \\ k \end{pmatrix}

Given that $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel, find the value of $k$ .

[4]

Question 10

HLPaper 1

Let $\vec{a}=\begin{pmatrix}2 \\ -1\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}4 \\ -2\end{pmatrix}$ .

Show that $\vec{a}$ and $\vec{b}$ are parallel.

[2]

AHL 3.10—Vector definitions Questionbank