Vector basics (position, displacement vectors, components ijk, rescaling/normalising vectors): notes and practice questions

Consider the points $A(1,2,5),\ B(7,3,0)$ and $C(4,6,2)$ and take all distances to be measured in meters.

$a$ Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

$a$ Find the area of triangle ABC.

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) 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) A satellite dish is supported by three anchor points A, B, and C on a mast. The coordinates of these points, relative to a fixed origin, are A(1, 2, 0), B(7, 2, 0), and C(4, 6, 1), where distances are measured in metres.

(i) Find the vector $\overrightarrow{\text{CA}}$.

(ii) Find the vector $\overrightarrow{\text{CB}}$.

(b) Calculate $\overrightarrow{\text{CA}} \times \overrightarrow{\text{CB}}$.

(c) Hence, determine the area of the triangular support surface formed by points A, B, and C.

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.

A small drone is tracking a moving target. Its position vector, $\vec{r}$, relative to a fixed origin $O$ at time $t$ (in seconds, $t \ge 0$) is given by

$\vec{r}(t) = \begin{pmatrix} t \cos(t) \\ t \sin(t) \end{pmatrix}$.

(a) Find the velocity vector of the drone, $\mathbf{v}(t)$.

(b) Show that the velocity vector of the drone is never parallel to its position vector for $t > 0$.

(a) The position vector of Drone A at time $t$ seconds is given by $\boldsymbol{r}_A = 4 \cos(3t)\boldsymbol{i} + 5 \sin(3t)\boldsymbol{j}$, where displacement is measured in metres.

(i) Find an expression for the velocity of Drone A at time $t$.

(ii) Hence, find the speed of Drone A when $t = 1.2$ seconds.

(b) (i) Find an expression for the acceleration of Drone A at time $t$.

(ii) Show that the acceleration of Drone A is always directed towards the origin.

(c) The position vector of a second drone, Drone B, is given by $\boldsymbol{r}_B = -5 \sin(4t)\boldsymbol{i} + 4 \cos(4t)\boldsymbol{j}$.

For $0 \le t \le 10$, find the time when the two drones are closest to each other.

(d) At time $k$, where $0 < k < 1.5$, Drone B is moving parallel to Drone A.

(i) Find the value of $k$.

(ii) At time $k$, show that the two drones are moving in the opposite direction.

In this question, $\mathbf{i}$ denotes a unit vector due east, and $\mathbf{j}$ denotes a unit vector due north.

Two drones, P and Q, are each flying with constant velocities.

The position vector of drone P, at time $t$ minutes, is given as $\mathbf{r}_P = (2 + 3t)\mathbf{i} + (5 - 2t)\mathbf{j}$.

The position vector of drone Q, at time $t$ minutes, is given as $\mathbf{r}_Q = (-1 + t)\mathbf{i} + (1 + 4t)\mathbf{j}$.

(a) Find the bearing on which drone P is flying.

(b) Find the value of $t$ when drone Q is directly west of drone P.

(c) Find the value of $t$ when drone Q is directly north-west of drone P.

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 triangular section of a modern architectural roof is being designed. The vertices of this section are represented by points P, Q, and R in a 3D coordinate system, where distances are measured in metres.

Point P is at $(0, 0, 2)$.

Point Q is at $(4, 7, 3)$.

Point R is at $(9, 2, 2)$.

Calculate the vector product $\vec{PQ} \times \vec{PR}$.

Hence, find the area of the triangular roof section.

A support beam is to be installed from point Q perpendicular to the edge PR. Find the length of this support beam.

The building's base lies on the horizontal plane (z=0). Find the acute angle that the roof section makes with the horizontal plane.

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 $a$.

(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 $S$.

(a) A ground crew on Luna Prime is tracking a supply drone. From the Aether station's control tower, the drone is observed to be $8500$ km East and $4000$ km North in a localized flat-map approximation.

(i) Find the straight-line distance from the Aether station to the drone.

(ii) Find the bearing of the drone from the Aether station. Give your answer in degrees, correct to one decimal place.

(b) The Aether station (A) is located at the origin $(0,0,0)$ of a 3D Cartesian coordinate system for this part. A research probe (P) is located at $(4000, 0, 0)$ km. A navigation beacon (B) is located at $(0, 4000, 0)$ km.

(i) Show that the position vector of the research probe, $\vec{p}$, is perpendicular to the position vector of the navigation beacon, $\vec{b}$.

(ii) Assuming the probe and beacon are on the surface of Luna Prime with a radius of $4000$ km, calculate the shortest distance along the surface between the research probe and the navigation beacon.

(c) Consider the Aether station (A) at $(4000, 0, 0)$ km, the Boreas station (B) at $(0, 4000, 0)$ km, and a North Pole reference point (N) at $(0, 0, 4000)$ km. Let $\vec{a}$, $\vec{b}$, and $\vec{n}$ be their respective position vectors from the centre of Luna Prime.

(i) Find the vector $\vec{a} \times \vec{b}$.

(ii) Show that the angle at vertex A in the spherical triangle formed by Aether, Boreas, and the North Pole (ABN) is $90^{\circ}$.

(d) A supply route between Aether and a new outpost, Delta, has an arc length of $2000$ km. Given that the radius of Luna Prime is $4000$ km, show that the central angle $\theta$ between Aether and Delta is $28.6^{\circ}$, correct to three significant figures.

(e) The Aether station (A) is located at $40^{\circ}$ N, $20^{\circ}$ E, and the Boreas station (B) is located at $70^{\circ}$ N, $100^{\circ}$ E on Luna Prime, which has a radius of $6000$ km. Find the shortest distance along the surface from Aether to Boreas.

(f) Using the vector method from part (c), find the initial bearing from Aether to Boreas. Give your answer in degrees, correct to one decimal place.

Three observation points in a 3D geological survey are located at points P, Q, and R. Their position vectors relative to an origin O are given by $\vec{OP} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$, $\vec{OQ} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}$ and $\vec{OR} = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}$.

(a) Calculate $\vec{PQ} \times \vec{PR}$.

(b) Hence find the area of triangle PQR.

A drone is programmed to fly from point $A(1, 2)$ to point $B(5, 5)$. During its flight, it experiences a constant wind force $\mathbf{W} = -10\mathbf{i} + 5\mathbf{j}$ Newtons.

Calculate the scalar component of the wind force $\mathbf{W}$ in the direction of the drone's intended path from $A$ to $B$.

Determine the scalar component of the wind force $\mathbf{W}$ which is perpendicular to the drone's intended path.

The wind force $\mathbf{W}$ can be resolved into two vector components: one acting in the direction of the drone's intended path and the other acting perpendicular to it.

State the component of the wind force $\mathbf{W}$ in the direction of the drone's intended path (from part (a) ) in vector form. Give your answer in terms of the unit vectors $\mathbf{i}$ and $\mathbf{j}$.

A deep-sea exploration probe is launched from a research vessel. Its velocity vector, $\mathbf{v}$, at time $t$ seconds after launch, is given by

$\mathbf{v} = \begin{pmatrix} 2t \\ 3t^2 - 1 \\ 4 \end{pmatrix}$ m/s.

At the moment of launch ($t = 0$), the probe's initial position relative to a fixed origin is $\mathbf{r}(0) = \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}$ m. Find the position vector, $\mathbf{r}(t)$, of the probe at time $t$.

Find the distance of the probe from the origin when $t = 2$ seconds.

(a) A deep-sea submersible is exploring an underwater trench. It is subjected to several forces, all measured in kN ($1$ kN $= 1000$ N).

Two main thrusters provide forces represented by the vectors $\begin{pmatrix} 45 \\ 10 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} 45 \\ -10 \\ 5 \end{pmatrix}$.

Two lateral thrusters exert forces of $\begin{pmatrix} 0 \\ 8 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ -8 \\ 0 \end{pmatrix}$.

The water resistance acting on the submersible is $\begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix}$ kN.

The buoyant force is $\begin{pmatrix} 0 \\ 0 \\ 120 \end{pmatrix}$ kN, and the weight of the submersible produces a force of $\begin{pmatrix} 0 \\ 0 \\ -100 \end{pmatrix}$ kN.

Find the resultant force acting on the submersible.

(b) Given that the mass of the submersible is $15\,000$ kg, find the acceleration of the submersible in ms$^{-2}$. The acceleration (ms$^{-2}$) of an object subject to a resultant force of $\mathbf{F}$ N is given by the formula $\mathbf{a} = \frac{\mathbf{F}}{m}$, where $m$ is the mass of the object in kilograms.

(c) The submersible is initially at the origin $(0,0,0)$ and has an initial velocity of $\begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}$ ms$^{-1}$.

(i) Find the displacement of the submersible at time $t$.

(ii) Find the displacement of the submersible at $t = 60$ s.

A robot arm applies a force $\mathbf{F}$ to move a component. The force vector is $\mathbf{F} = \begin{pmatrix} 7 \\ -3 \end{pmatrix}$ newtons (N). The component is moved from an initial position $P_1(2, 5)$ to a final position $P_2(8, 1)$, where coordinates are in metres. Calculate the work done by the robot arm.

In physics, the work done $W$ by a constant force $\mathbf{F}$ moving an object through a displacement $\vec{d}$ is generally given by the scalar product $W = \mathbf{F} \cdot \vec{d}$. Explain why, if the force is applied exactly in the direction of the displacement, the formula simplifies to $W = |\mathbf{F}| |\vec{d}|$.

A deep-sea submersible starts its descent from an initial position $Q_1(10, 20, 5)$ metres and travels in a straight line to a target location $Q_2(70, 60, 25)$ metres. The coordinates are relative to a fixed origin on the surface. The submersible's propulsion system generates a constant force of $150\,000$ N in the direction of motion. Using the simplified formula from part (b), calculate the total energy expended by the propulsion system during this descent. Give your answer to three significant figures.