Two autonomous vehicles move for $0\le t\le5$, where $t$ is measured in minutes. Their position vectors, in kilometres, are

Two lines have vector equations

Two vehicles start moving at the same time. Their positions after $t$ hours are

Distances are measured in kilometres.

The position vector of a particle moving in a horizontal plane is

where $t$ is measured in seconds.

A force $\boldsymbol{F}=\begin{pmatrix}0\\40\\-30\end{pmatrix}$ newtons is applied at a point whose position relative to a pivot is $\boldsymbol{r}=\begin{pmatrix}0.25\\0.10\\0\end{pmatrix}$ metres. The torque vector is defined by $\boldsymbol{\tau}=\boldsymbol{r}\times\boldsymbol{F}$.

A straight pipeline is modelled by the line

A monitoring station is located at $P(7,3,1)$.

A particle $A$ moves according to

where $t$ is measured in seconds. A second particle $B$ follows the same motion exactly $2$ seconds later.

An aircraft flies in a horizontal plane. East and north are represented by the positive $x$- and $y$-directions respectively. The aircraft has an airspeed of $240\ \text{km h}^{-1}$ in the direction $3\boldsymbol{i}+4\boldsymbol{j}$. A constant wind has velocity $-30\boldsymbol{i}+20\boldsymbol{j}\ \text{km h}^{-1}$.

A straight railway tunnel passes through the points $A(-2,1,4)$ and $B(4,5,-2)$. A maintenance shaft is modelled by the line

A towing cable exerts a force of magnitude $500\ \text{N}$ in the direction $4\boldsymbol{i}+\boldsymbol{j}$. During one stage of the tow, the load has displacement $24\boldsymbol{i}+10\boldsymbol{j}$ metres.

Three environmental sensors are placed at $A(2,-1,3)$, $B(8,2,0)$ and $C(-1,5,6)$. A communications hub is to be positioned using the medians of triangle $ABC$.

A robotic tool applies the constant force $\boldsymbol{F}=(3,-2,1)$ newtons while moving from $O=(0,0,0)$ to $B=(4,1,3)$ metres. It may move directly or via $A=(1,3,-1)$.

Two survey vessels move for $0\leq t\leq6$, where $t$ is measured in hours and positions are measured in kilometres:

A water droplet moves in a vertical plane. Its velocity at time $t$ seconds is

At $t=0$, its position is $(-3,2)$ metres.

A rectangular solar panel has adjacent vertices $A(1,2,0)$, $B(7,3,-2)$ and $D(2,6,5)$. Coordinates are measured in metres. The direction from the panel towards the Sun is represented by $\boldsymbol{s}=\begin{pmatrix}2\\-1\\-2\end{pmatrix}$.

A mechanic applies a force of magnitude $180\ \text{N}$ in the direction $3\boldsymbol{j}-4\boldsymbol{k}$. The position vector from the centre of a bolt to the point of application is

The torque vector is $\boldsymbol{\tau}=\boldsymbol{r}\times\boldsymbol{F}$.

In a computer-graphics model, a surface has outward normal vector $\boldsymbol{n}=2\boldsymbol{i}-\boldsymbol{j}+2\boldsymbol{k}$. The direction from the surface towards a light source is $\boldsymbol{s}=\boldsymbol{i}-2\boldsymbol{j}+2\boldsymbol{k}$. The illumination factor is modelled by the cosine of the acute angle between the two vectors.

A river is $600\ \text{m}$ wide and flows due east at $1.8\ \text{m s}^{-1}$. A boat moves through the water at a constant speed of $4.5\ \text{m s}^{-1}$. The boat starts on the south bank directly opposite a landing point on the north bank. The diagram is schematic and not to scale; the boat's heading is specified in each part.

A particle $A$ moves around a circle according to

A particle $B$ follows the same motion exactly $4$ seconds later.

Two inspection pods move through a three-dimensional storage facility. Their positions after $t$ seconds are

Distances are measured in metres.

Two cables exert forces on a joint. Cable 1 exerts a force of magnitude $8\ \text{kN}$ in the direction $2\boldsymbol{i}-\boldsymbol{j}+2\boldsymbol{k}$. Cable 2 exerts a force of magnitude $10\ \text{kN}$ in the direction $-3\boldsymbol{i}+4\boldsymbol{j}$. A mast through the joint has direction $\boldsymbol{j}+\boldsymbol{k}$.

A stationary camera drone is at $D(4,-3,12)$ metres. Its camera is initially directed towards a ground marker $M(10,5,0)$. The ground is represented by the plane $z=0$.

An underwater vehicle travels from $A(1,2,-1)$ to $B(13,8,7)$. Coordinates are measured in metres. The journey takes $6$ minutes. The water current has velocity

in metres per minute. The vehicle maintains a constant velocity relative to the water.

Two straight mining tunnels are modelled by

Two aircraft fly at constant altitude. Their positions, in kilometres, are

where $t$ is measured in minutes and $0\leq t\leq4$. A separation below $2$ km requires an alert.

Two inspection robots follow straight routes across a factory floor. Robot $A$ starts at time $t=0$ and has position

Robot $B$ begins at $(8,0)$ and moves with velocity $\begin{pmatrix}-1\\2\end{pmatrix}$ once activated.

A camera moves around an elliptical track. Its position after $t$ seconds is

in metres. For part (b), take the first circuit to mean $0\leq t<4\pi$, with $t$ measured in seconds.

A parallelogram-shaped solar panel has adjacent edge vectors

in metres. Sunlight travels parallel to $\boldsymbol{s}=(2,-1,2)$.

At an airport, the wind velocity is $\boldsymbol{w}=(8,-6,2)\ \text{m s}^{-1}$. A sloping runway has direction vector $\boldsymbol{d}=(2,1,2)$ in the stated take-off direction.

A triangular fabric panel has vertices $A=(0,0,0)$, $B=(4,0,0)$ and $C=(1,k,2)$, where $k$ is a real design parameter. Coordinates are measured in metres.