AHL 3.11—Vector equation of a line in 2d and 3d Questionbank

Practice AHL 3.11—Vector equation of a line in 2d and 3d 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

A line L has the vector equation

\vec{r}=\left[\begin{array}{c} 1 \\ 2 \\ -3 \end{array}\right]+\lambda\left[\begin{array}{c} 2 \\ -1 \\ 4 \end{array}\right]

Find the coordinates of the point on line L that is closest to the point $\mathrm{A}=(3$ , $-2,1)$ .

[6]

Question 2

HLPaper 1

A line L is defined by the vector equation

\vec{r}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right]

Find the value of $\lambda$ forwhichthepointonthelineliesontheplanex $+y+z=5$ .

[4]

Question 3

HLPaper 1

A line passes through the point $P(4,-1,2)$ and is parallel to the vector $\left[\begin{array}{c}2 \\ 0 \\ -3\end{array}\right]$ .

Write down the vector equation of the line.

[3]

Question 4

HLPaper 1

The line $\mathrm{L}_{1}$ isgivenby $\vec{r}=\left[\begin{array}{l}1 \\ 3 \\ 2\end{array}\right]+\lambda\left[\begin{array}{c}2 \\ -1 \\ 1\end{array}\right]$ and point $\mathrm{A}=(5,1,3)$ .

Find the shortest distance from point A to line $\mathrm{L}_{1}$ .

[5]

Question 5

HLPaper 1

A line in 3D passes through the points $P(0,1,-2)$ and $Q(2,3,0)$ .

Find its vector equation in parametric form.

[4]

Question 6

SLPaper 1

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

The line through A and B is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .

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

Question 7

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while anarcher attempts to hit it with an arrow.

The path of the ball is modelled by the equation

$(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 5 \\ 0 \end{matrix}) + t (\begin{matrix} u_{x} \\ u_{y} - 5 t \end{matrix})$

where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacementfrom the ground, both measured in metres, and $t$ is the time, in seconds, since the ballwas 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 launchedwith an initial velocity such that $u_{x} = 8$ and $u_{y} = 10$ .

An archer releases an arrow from the point $(0, 2)$ . The arrow is modelled as travelling in astraight line, in the same plane as the ball, with speed $60 {m s}^{- 1}$ and an angle of elevation of $10 °$ .

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]

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 ballreaches its maximum height.

[4]

Question 8

HLPaper 2

Consider a vector line in 2D represented by the vector equation $\mathbf{r} = \mathbf{a} + t\mathbf{b}$ , where $\mathbf{a}$ is a position vector and $\mathbf{b}$ is a direction vector.

Given $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ , find the coordinates of the point on the line when $t = 2$ .

[2]

Determine the equation of the line in the form $y = mx + c$ .

[3]

Question 9

HLPaper 1

Two lines L₁ and L₂ are given by the following equations, where p ∈ ℝ. L₁: r = (2, p+9, -3) + λ(p, 2p, 4) L₂: r = (14, 7, p+12) + μ(p+4, 4, -7) It is known that L₁ and L₂ are perpendicular.

Find the possible value(s) for p.

[3]

In the case that p < 0, determine whether the lines intersect.

[4]

Question 10

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]

At an archery tournament, a particular competition sees a ball launched into the air while anarcher attempts to hit it with an arrow.

The path of the ball is modelled by the equation

$(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 5 \\ 0 \end{matrix}) + t (\begin{matrix} u_{x} \\ u_{y} - 5 t \end{matrix})$

where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacementfrom the ground, both measured in metres, and $t$ is the time, in seconds, since the ballwas 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 launchedwith an initial velocity such that $u_{x} = 8$ and $u_{y} = 10$ .