Distance between two points, midpoint (up to 3d), angle between two lines: notes and practice questions

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.

Three emergency service hubs are located at coordinates F1 (2, 5), P1 (10, 5), and H1 (10, 15), where distances are measured in kilometers. A central dispatch unit needs to be built at a location equidistant from these three hubs. This location is a vertex of the Voronoi diagram formed by the hubs.

Write down the coordinates of this central dispatch unit.

Find the equation of the perpendicular bisector of the segment connecting F1 (2, 5) and H1 (10, 15).

A new emergency hub, S1 (4, 8), is established, and the Voronoi diagram is redrawn. Several potential locations for a new drone charging station are at the vertices of this updated diagram. The coordinates of four such vertices are given as V_alpha (6, 10), V_beta (6, 4.5), V_gamma (8.75, 10), and V_delta ($\frac{223}{28}$, $\frac{59}{7}$). The drone charging station should be built at the vertex that is as far as possible from its nearest emergency service hub.

By calculating appropriate distances, find the location of the drone charging station.

Hence, write down the distance of the drone charging station to the nearest emergency service hub.

A team of geologists is mapping underground rock formations using a 3D coordinate system. The x and y axes represent horizontal distances across the surface, and the z-axis represents depth below the surface (with positive z indicating deeper). However, for this particular survey, the z-axis is set to represent altitude above a reference point, so positive z means higher.

Two key geological markers, P and Q, have been identified. Marker P is located at coordinates (100, 20, 50) and Marker Q is at (40, 60, 290). All coordinates are given in metres.

(a) Find the straight-line distance between Marker P and Marker Q.

A new sensor, S, is to be installed exactly halfway between Marker P and Marker Q.

(b) Find the coordinates of the sensor S.

(c) Write down the altitude of the sensor S, in metres, above the reference point.

A botanical garden features a winding path for visitors. The path can be modelled by the function $f(x) = x^3 - 3x^2 - 9x + 5$. All distances in the garden are in kilometres.

A new straight maintenance path needs to be constructed. This path will start at point P on the visitor path, which has coordinates $(2, -17)$.

(a) Using your graphic display calculator, find the value of $f'(2)$.

(b) Find the equation of the line normal to $f(x)$ at point P.

(c) The maintenance path connects point P to another point Q on the visitor path, where the normal line intersects the path again. For safety regulations, point Q must have a positive x-coordinate. Determine the length of this new maintenance path.

A drone takes off from a launch pad located at the origin O(0, 0, 0) of a coordinate system. It flies in a straight line to a target destination T with coordinates (5.0, 3.0, 7.0). All units are in metres.

(a) Calculate the total distance the drone travels from the launch pad to the target.

(b) Determine the angle of elevation of the drone's flight path from the horizontal ground.

A drone is used for aerial surveying. It starts at a central hub (H).

(a)(i) From H, it flies 15 km on a bearing of 040° to survey point S1. From S1, it flies 10 km on a bearing of 170° to survey point S2.

Determine the interior angle HS1S2.

(a)(ii) Determine the distance from S2 back to H.

(a)(iii) Determine the bearing the drone must use to travel directly from S2 to H.

(b) The drone's mission also includes a final delivery point D, such that S1S2DH forms a parallelogram. Write down the distance between point D and point H. Justify your answer.

Two historical landmarks are located at points A(4, 10) and B(16, 2) on a coordinate map. A proposed new scenic path, represented by the line R with equation $y = x - 2$, passes near these landmarks. A new visitor centre is to be built on this scenic path such that it is equidistant from both landmarks.

(a) Find the equation of the perpendicular bisector of the line segment [AB]. Give your equation in the form $y = mx + c$.

(b) Determine the coordinates of the point on the scenic path R where the visitor centre should be located.

(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.

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) 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.

Three environmental sensor stations are located at S1(15, 40), S2(5, 25), and S3(30, 30) according to a local coordinate system, where distances are measured in kilometers.

A new construction site is planned at position C(22, 35).

Find the distance between the construction site C and sensor station S1.

On a particular day, the mean PM2.5 levels recorded at sensor stations S1, S2, and S3 are 28 $\mu$g/m$^3$, 35 $\mu$g/m$^3$, and 20 $\mu$g/m$^3$ respectively.

Use nearest neighbour interpolation to estimate the PM2.5 level at the construction site C on that particular day.

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.

Relative to a central command station, two space probes, P and Q, have positions $(3.5, 6.2, 1.8)$ and $(2.1, 7.5, 2.3)$ respectively, where the units of measurement are in millions of kilometres.

Find the distance between the two space probes.

Determine which of the space probes is closer to the central command station.

A geological survey team is setting up sensors in a remote area. A map of the area is represented on the following coordinate axes.

Three sensor locations are positioned at $S_1(0, 10)$, $S_2(16, 14)$ and $S_3(16, 0)$.

All measurements are in kilometres.

(a) Write down the distance between $S_2$ and $S_3$.

(b) Calculate the distance between $S_1$ and $S_2$.

(c) A geological team member is at sensor $S_2$ and needs to walk directly to sensor $S_1$. Calculate the bearing of $S_1$ from $S_2$.

A communication relay station is to be installed at a point that is an equal distance from each of the sensors at $S_1$, $S_2$, and $S_3$.

(i) Write down the gradient of the line segment $[S_1S_3]$.

(ii) Write down the coordinates of the midpoint of the line segment $[S_1S_3]$.

(iii) Hence, calculate the coordinates of the communication relay station.

A drone is flying between two observation towers. Tower A is located at coordinates $(-4, 10)$ and Tower B is located at coordinates $(6, 2)$. The drone's flight path is a straight line segment AB. The drone needs to refuel at a station located exactly halfway along its flight path.

Find the coordinates of the refueling station.

Let $L_1$ be the line representing the drone's flight path from Tower A to Tower B.

Find the gradient of $L_1$.

A security laser beam, $L_2$, is emitted from the refueling station and is designed to be perpendicular to the drone's flight path.

Write down the gradient of $L_2$.

Find the equation of $L_2$. Give your answer in the form $y = mx + c$.

A drone delivery service operates from three main hubs, P, Q, and R, whose locations are represented on a coordinate plane. All measurements are in kilometres.

Hub P is at $(0, 10)$, Hub Q is at $(8, 14)$, and Hub R is at $(8, 2)$.

Write down the distance between Hub Q and Hub R.

Calculate the distance between Hub P and Hub Q.

A drone is at Hub Q and needs to fly directly to Hub P. Calculate the bearing of P from Q.

A central charging station is to be built at a location equidistant from Hubs P, Q, and R.

Write down the gradient of the line segment [PR].

Find the coordinates of the midpoint of [PR].

Hence calculate the coordinates of the central charging station.

A geological survey team is establishing a network of sensor stations in a remote volcanic region. These stations are connected by automated drone routes for data collection. The coordinates of the stations are given in kilometres relative to a central base camp.

The coordinates of the Base Camp (A) are $(0, 0, 0)$.

Observation Post 1 (B) is located at $(600, 300, 200)$.

Observation Post 2 (C) is located at $(1000, 800, 600)$.

(a) Determine the distance a drone travels from the Base Camp (A) to Observation Post 1 (B).

(b) For a drone to reach Observation Post 2 (C), it must first travel from the Base Camp (A) to Observation Post 1 (B), and then take a separate route from B to C.

Determine the total distance covered by a drone travelling from the Base Camp (A) to Observation Post 2 (C) via Observation Post 1 (B). Give your answer to three significant figures.

The 'EcoPack' company designs sustainable packaging. They produce a standard closed rectangular storage container with a length of $10$ cm, a width of $6$ cm, and a height of $4$ cm. The information is shown in the diagram.

Calculate the surface area of the container in cm$^2$.

(b) Calculate the length of the longest internal diagonal of the container.

(c) Each week, EcoPack sells $x$ thousand containers. It is known that $\frac{dP}{dx} = -3x+300$, for $x \ge 0$, where $P$ is the weekly profit, in dollars, from the sale of $x$ thousand containers.

Find the number of containers that should be sold each week to maximize the profit.

(d) The profit from the sale of $30000$ containers is $2500.

Find $P(x)$.

(e) Find the least number of containers which must be sold each week in order to make a profit.

A remote research station, R, located at coordinates $(100, 150)$, requires supplies to be dropped off at a point along a straight supply route, S. The supply route S can be modelled by the equation $y = -x + 200$. All coordinates are given in kilometres.

To minimize travel time for the delivery drone, the drop-off point, D, must be the point on the supply route closest to the research station.

Determine the coordinates of D, the optimal supply drop-off point.

Find the distance between the research station R and the optimal drop-off point 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 $a$.