Let

where $k>0$. The angle between $\mathbf{u}$ and $\mathbf{v}$ is $1.10$ radians.

The line $L_1$ passes through the points $A(1,2,-1)$ and $B(4,-2,5)$. The line $L_2$ has equation

The lines $L_1$ and $L_2$ are given by

The points $A$, $B$ and $C$ have coordinates $A(2,-1,0)$, $B(5,1,4)$ and $C(0,3,2)$.

A plane $\i$ has vector equation

The point $P(k,1,0)$ lies on $\i$.

The line $L$ and plane $\pi$ are given by

Consider the lines

where $\lambda,\mu,k\in\mathbb{R}$.

The plane $\Pi$ passes through the points $A(1,0,-1)$, $B(2,3,0)$ and $C(-1,1,2)$.

The planes $\Pi_1$ and $\Pi_2$ are given by

where $p\in\mathbb{R}$. The acute angle between the planes is $\dfrac{\pi}{4}$.

The line $L$ has equation

The point $P$ has coordinates $(0,3,1)$. A plane $\Pi$ contains $L$ and the point $P$.

The planes $\Pi_1$ and $\Pi_2$ are given by

The planes intersect in the line $L$.

Two drones, $A$ and $B$, move in straight lines. Their position vectors, in kilometres, $t$ hours after 09:00 are

where $0\leq t\leq 4$.

The planes $\i_1$ and $\i_2$ are given by

where $k>1$. The acute angle between the planes is $1.10$ radians.

The planes $\i_1$ and $\i_2$ are given by

The planes intersect in the line $L$.

The points $A(1,2,0)$, $B(3,0,1)$ and $C(0,1,4)$ lie in the plane $\Pi$. A point $D$ lies on the line through $A$ perpendicular to $\Pi$.

The line $L$ has equation $\mathbf r=\begin{pmatrix}2\\-1\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-2\end{pmatrix}$, $\lambda\in\mathbb R$. The point $P$ has coordinates $(5,0,1)$.

The lines $L_1$ and $L_2$ are defined by $L_1:\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}$ and $L_2:\mathbf r=\begin{pmatrix}3\\k\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-1\\2\end{pmatrix}$, where $k\in\mathbb R$.

Two lines are given by $L_1:\mathbf r=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-1\end{pmatrix}$ and $L_2:\mathbf r=\begin{pmatrix}1\\0\\3\end{pmatrix}+\mu\begin{pmatrix}1\\2\\-1\end{pmatrix}$, where $\lambda,\mu\in\mathbb R$.

Two inspection robots, $A$ and $B$, move along straight paths inside a storage facility. Their position vectors, in metres, $t$ minutes after 10:00, are

where $0\leq t\leq 4$.

A triangular metal plate has vertices $A(1,2,-1)$, $B(5,0,2)$ and $C(2,6,1)$. A laser path is modelled by the infinite line

The line $L$ is given by

A plane $\pi$ contains $L$ and the point $P(4,0,5)$.

A navigation system records three points in space, $A(-6,2,-1)$, $B(0,1,1)$ and $C(-4,6,-2)$. A fourth point $D$ is to be chosen so that $ABDC$ is a parallelogram, with $AB$ and $AC$ as adjacent sides.

Two particles $A$ and $B$ move in straight lines. At time $t$ seconds, their position vectors are $\mathbf r_A=\begin{pmatrix}1\\2\\0\end{pmatrix}+t\begin{pmatrix}2\\-1\\1\end{pmatrix}$ and $\mathbf r_B=\begin{pmatrix}5\\1\\0\end{pmatrix}+t\begin{pmatrix}-1\\1\\0\end{pmatrix}$, where $t\geq0$.

The planes $\Pi_1$ and $\Pi_2$ have equations $\Pi_1:x+2y-z=4$ and $\Pi_2:2x-y+z=0$. Their line of intersection is $L$.

The plane $\Pi$ has vector equation $\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+s\begin{pmatrix}1\\1\\-1\end{pmatrix}+t\begin{pmatrix}2\\-1\\1\end{pmatrix}$, where $s,t\in\mathbb R$. The line $L$ is given by $\mathbf r=\begin{pmatrix}3\\-2\\5\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\-1\end{pmatrix}$.