Vector equations of a line in 2D & 3D (different forms too): notes and practice questions

Consider line $L_{1}$ with the vector equation $r = \begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix} + t\begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}$.

$a$ Check if point A$(4,11,14)$ passes through line $L_{1}$.

Consider line $L_{2}$ with the vector equation $r = \begin{pmatrix} 2 \\ 3 \\ - 4 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ 4 \\ 9 \end{pmatrix}$.

$b$ Determine a vector perpendicular to both lines $L_{1}$ and $L_{2}$.

A high-tech drone is launched from a control tower at coordinates (1.5, 2.0, 0.8) relative to the base of the tower 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 drone travels with a constant velocity, and its direction of travel is given by the vector $\begin{pmatrix} -1 \\ -2 \\ -0.4 \end{pmatrix}$.

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

The drone is programmed to land on a designated pad at ground level (where $z=0$ ).

(i) Find the value of the parameter $\lambda$ when the drone reaches ground level.

The drone is programmed to land on a designated pad at ground level (where $z=0$ ).

(ii) Determine the coordinates of the landing pad.

Calculate the distance the drone travels from its initial position to the landing pad.

(a) A space probe, 'Voyager Alpha', is launched from a space station. The position of the probe at time $t$ days after launch is given by the vector

$\mathbf{r} = \begin{pmatrix} 10 \\ 20 \\ 5 \end{pmatrix} + t \begin{pmatrix} 80 \\ 40 \\ 10 \end{pmatrix}$

Distances are measured in thousands of kilometres.

Find the position vector of the probe $3$ days after launch.

(b) A second space probe, 'Explorer Beta', is launched from a different station. The position vector of this probe is given by the vector

$\mathbf{s} = \begin{pmatrix} -50 \\ -30 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 70 \\ 50 \\ 12 \end{pmatrix}$

Determine if the two flight paths intersect and, if so, state the point of intersection.

(c) The two probes were launched at the same time, so $\lambda = t$.

State, with a reason, whether the two probes actually collide.

(d) Calculate the distance between the two space stations (the initial launch points).

(e) Calculate the shortest distance that ever exists between the two probes and the time when this occurs. Assume $t \ge 0$ days.

The paths of two automated guided vehicles (AGVs), $L_1$ and $L_2$, in a large warehouse are modelled by the following vector equations, where $k\in\mathbb{R}$ is a constant:

$L_1:\boldsymbol{r}=\begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} + \lambda \begin{pmatrix} k \ 5 \ -6 \end{pmatrix}$

$L_2: \boldsymbol{r} = \begin{pmatrix} 10 \ -1 \ 8 \end{pmatrix} + \mu \begin{pmatrix} k+2 \ 3 \ 5 \end{pmatrix}$

It is known that the paths of the AGVs are perpendicular.

(a) Find the possible value(s) for $k$.

(b) In the case that $k < 0$, determine whether the lines intersect.

(a) The position of a reconnaissance drone, relative to a control tower, is given by the vector equation $r = \begin{pmatrix} 7 \\ 2 \end{pmatrix} + t \begin{pmatrix} -3 \\ 4 \end{pmatrix}$, where $r$ is the position vector in metres and $t$ is the time in minutes.

Write down the position vector of the drone when $t = 0$ and when $t = 1$.

(b) Calculate the speed of the drone.

(c) Find an expression for the distance of the drone from the origin at time $t$.

(d) Hence find the minimum distance of the drone from the origin and the time at which it occurs.

A surveillance drone D is flying with a constant velocity, $\boldsymbol{v}$, measured in kilometres per hour, where

$\boldsymbol{v} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}$.

At time $t = 0$ the drone is at a point A(50, 20) relative to an origin O, where distances are

measured in kilometres.

Find the position vector $\vec{OD}$ of the drone at time $t$ hours.

A protected bird's nest is located at a point N(99, 2).

Find the value of $t$ when the drone will be closest to the bird's nest.

An environmental sensor will trigger if the drone flies within 18 kilometres of the nest.

State whether the sensor will trigger. Give a reason for your answer.

A drone A takes off from a control tower at 10:00. It flies north-east at a horizontal speed of $70\sqrt{2} \text{ kmh}^{-1}$ and climbs at a rate of $3 \text{ kmh}^{-1}$. At 10:00, it is at a height of $5 \text{ km}$ directly above the control tower.

Find an expression for the displacement of drone A from the control tower at time $t$ hours after 10:00. Assume the control tower is at the origin (0,0,0) and the positive y-axis points North, and the positive x-axis points East.

At 10:30, a second drone B is $12 \text{ km}$ directly above the control tower. It flies on a bearing of $300^\circ$ at a horizontal speed of $80 \text{ kmh}^{-1}$ and descends at a rate of $2 \text{ kmh}^{-1}$.

Find an expression for the displacement of drone B from the control tower $t$ hours after 10:00.

Find the distance the two drones are apart when they have the same height.

At 10:00 am, a reconnaissance drone is located 2 km East and 6 km North of a central control tower. A coordinate system is established with the control tower at the origin. The drone maintains a constant velocity of $\begin{pmatrix} -0.8 \\ -1.2 \end{pmatrix}$ kilometres per hour (km h$^{-1}$), where the components represent velocity in the East-West and North-South directions, respectively.

Write down an expression for the position vector $\mathbf{r}$ of the drone, $t$ hours after 10:00 am.

Find the time at which the bearing of the drone from the control tower is 180°.

The position of a drone, D$_1$, $t$ seconds after leaving a control tower T, is given by

$\mathbf{r} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix} + t \begin{pmatrix} -2 \\ 3 \\ 4 \end{pmatrix}$, $t \ge 0$.

The units of distance are metres.

Write down the coordinates of the control tower T.

Four seconds after leaving T, D$_1$ is at point P.

Find the displacement vector $\vec{\text{TP}}$.

Find the distance $|\vec{\text{TP}}|$.

A second drone, D$_2$, leaves the control tower T at the same time as D$_1$. D$_2$ is moving in the direction of the vector $\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$.

Find the angle between the initial flight paths of Drone 1 and Drone 2.

The drone D$_2$ has a speed of $10$ m s$^{-1}$.

Find the distance between Drone 1 and Drone 2 when $t = 4$ seconds.

A civil engineer is designing a new road. A preliminary section of the road is modelled by a straight line with equation $y = kx + d$.

Find the vectors $\mathbf{a}$ and $\mathbf{b}$ such that the equation of the line can be expressed in vector form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$ in terms of $k$ and/or $d$.

Before construction, a ground transformation is applied to the design. This transformation is described by the matrix $T = \begin{pmatrix} 4 & 2 \\ 6 & 3 \end{pmatrix}$.

Calculate the value of $\det T$.

The preliminary road section $y = kx + d$ (where $k \neq -2$) undergoes the transformation described by matrix $T$.

Show that the equation of the resulting transformed path does not depend on $k$ or $d$.

A deep-sea research submersible, 'Nautilus', is being tracked relative to an underwater research station, 'Triton Base'. The coordinates $(x, y, z)$ represent the submersible's displacement in kilometres, where $x$ is east, $y$ is north, and $z$ is vertical displacement (positive upwards, so negative for depths below sea level).

At 10:00 AM, the submersible is detected at a position 60 km east and 24 km north of Triton Base, and at a depth of 15 km below sea level. Its velocity is given as $\begin{pmatrix} -120 \\ -48 \\ -10 \end{pmatrix}$ kmh$^{-1}$. Let $t$ be the length of time in hours from 10:00 AM.

Write down a vector equation for the displacement, $\vec{r}$, of the submersible in terms of $t$.

If the submersible continued to travel with the given velocity,

verify that it would pass directly over Triton Base (the point $(0,0,0)$);

state the depth of the submersible at this point;

find the time at which it would pass directly over Triton Base.

When the submersible is at a depth of 18 km below sea level, it continues to move horizontally on the same bearing but adjusts its vertical velocity so that it will dock precisely at Triton Base $(0,0,0)$.

Find the time at which the submersible is at a depth of 18 km below sea level.

Find the direct distance of the submersible from Triton Base at this point.

Given that the velocity of the submersible, after the adjustment of the vertical velocity, is $\begin{pmatrix} -120 \\ -48 \\ a \end{pmatrix}$ kmh$^{-1}$, find the value of $a$.

The flight path of drone A can be modelled by the vector equation $\boldsymbol{r}_A = \begin{pmatrix} 1 \\ 5 \\ -2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$, where $s \in \mathbb{R}$ represents time in minutes.

The flight path of drone B can be modelled by the vector equation $\boldsymbol{r}_B = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$, where $t \in \mathbb{R}$ represents time in minutes.

The two drones intersect at a point P. Find the coordinates of P.

Find the acute angle between the flight paths of the two drones.

A robotics engineer is programming a swarm of drones. The initial paths of the drones are given by a family of lines, $L_k$, with equation $y = kx + (5-2k)$, where $k$ is a real parameter and $k \neq -1.5$.

A control signal applies a linear transformation to the coordinates of every drone, described by the matrix $T = \begin{pmatrix} 3 & 2 \\ -6 & -4 \end{pmatrix}$.

The new path of a drone is the line $L'_k$.

(a) Find a vector equation for the line $L_k$ in terms of $k$.

(b) Find the determinant of $T$.

(c) Show that all the drones will end up moving along the same path, by showing that the equation of the transformed line $L'_k$ does not depend on $k$.

(a) A drone is programmed to fly along a straight path $L$. Its position at time $t$ (in minutes) can be modelled by the vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$, where the coordinates are in metres. A stationary sensor $S$ is located at the point $(10, 4, 4)$.

Find the coordinates of the point $P$ on the drone's path that is closest to the sensor $S$.

(b) Find a vector that is perpendicular to both the drone's path $L$ and the line segment connecting point $P$ to the sensor $S$.

Two drones, Alpha and Beta, are flying in a designated airspace. Their positions at time $t$ hours, $0 \le t < 15$, are given by the position vectors $r_A = \begin{pmatrix} 10 \\ 5 \\ 2 \end{pmatrix} + t \begin{pmatrix} -1 \\ -0.5 \\ 0.2 \end{pmatrix}$ and $r_B = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 0.6 \\ 0.3 \\ 0.4 \end{pmatrix}$ respectively, relative to a control tower (all lengths are in kilometres).

Show that the two drones would collide at a point P and write down the coordinates of P.

To avoid a collision, Drone Beta adjusts its velocity so that its position vector is now given by $r_B = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 0.3 \\ 0.15 \\ 0.2 \end{pmatrix}$.

Find the value of $t$ when Drone Beta, with its adjusted path, passes through point P.

Find the value of $t$ when the two drones (Drone Alpha and the adjusted Drone Beta) are closest together.

Find the distance between the two drones at this time.

A drone, 'SkyRanger 1', is flying along a path that can be modelled by the line $L_1$ with equation $\mathbf{r} = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix} + t \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix}$, where $t$ is a time parameter in minutes. A ground control station detects SkyRanger 1 at a specific point A with coordinates $(-7, 11, k)$.

(a) Calculate the value of $k$.

A second drone, 'AeroScout 2', is dispatched. Its flight path is modelled by the line $L_2$ with equation $\mathbf{r} = \begin{pmatrix} -15 \\ 7 \\ u \end{pmatrix} + s \begin{pmatrix} 2 \\ p \\ q \end{pmatrix}$, where $s$ is a time parameter in minutes. AeroScout 2 is designed to intercept SkyRanger 1 at point A and its path $L_2$ is perpendicular to the path of SkyRanger 1.

(b) Determine the values of $p$, $q$, and $u$.

A rectangular prism (cuboid) is used as a building block. It has one vertex at the origin O(0,0,0). Its dimensions are 6 units along the x-axis, 4 units along the y-axis, and 3 units along the z-axis. The vertices are given by O(0,0,0), A(6,0,0), B(6,4,0), C(0,4,0), D(0,0,3), E(6,0,3), F(6,4,3), and G(0,4,3).

Find the surface area of the cuboid.

Find the length of the space diagonal [OF].

The space diagonals [OF] and [CE] intersect at the point M.

Find the coordinates of M.

Find the acute angle between the diagonals [OF] and [CE] at their intersection point M.

(a) A reconnaissance drone is launched from a base station at point $A(5, 2, 1)$ relative to an origin $O$, where lengths are measured in metres and time, $t$, is measured in seconds. It flies in a straight line with vector equation

$\vec{r} = \begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix} + t \begin{pmatrix} 4 \\ 4 \\ 2 \end{pmatrix}$

Find the speed of the reconnaissance drone (in m/s).

(b) A patrol drone is dispatched from a different location with position vector $\begin{pmatrix} -4 \\ 29 \\ 37 \end{pmatrix}$, relative to $O$. It flies with a constant velocity vector of $\begin{pmatrix} 7 \\ -5 \\ -10 \end{pmatrix}$ to intercept the reconnaissance drone.

Write down the vector equation for $\vec{p}$, that models the flight of the patrol drone.

(c) Find the position vector at which the patrol drone intercepts the reconnaissance drone.

A drone is being tracked in a 3D coordinate system. At time $t_1$, its position is at point A$(1, -2, 3)$. At a later time $t_2$, its position is at point B$(5, 4, -1)$.

Find the distance the drone travelled from A to B.

Another drone's flight path is modelled by the line $L$ with vector equation $\boldsymbol{r} = \begin{pmatrix} 0 \\ 1 \\ 5 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \\ -2 \end{pmatrix}$. A control tower observes this drone at point C$(6, y, -1))$ which lies on line $L$.

Find the value of $y$.

Using the value of $y$ found in part (b), the position of point C is $(6, 10, -1))$.

Show that $\vec{AC} = \begin{pmatrix} 5 \\ 12 \\ -4 \end{pmatrix}$.

Find the angle between the drone's flight segment $\vec{AB}$ and the segment $\vec{AC}$. Give your answer in degrees to one decimal place.

Find the area of the triangle formed by the points A, B, and C. Give your answer to three significant figures.