Vector Properties (DP IB Analysis & Approaches (AA): HL): Exam Questions

Consider the two vectors $\mathit{s}=3\mathit{i}+4\mathit{j}-\mathit{k}$ and $\mathit{t}=-2\mathit{i}+2\mathit{j}-3\mathit{k}$.

(i) Find the cross product of $\mathit{s}$ and $\mathit{t}$.

(ii) Hence, find the angle between $\mathit{s}$ and $\mathit{t}$. Give your answer in radians.

The vectors $\mathit{a}$ and $\mathit{b}$ are defined by $\mathit{a}=\left(\begin{array}{c}1\\ -3\\ 1\end{array}\right)$ and $\mathit{b}=\left(\begin{array}{c}5\\ 2\\ -2\end{array}\right)$.

By finding the scalar product of $\mathit{a}$ and $\mathit{b}$, find the angle between them. Give your answer in degrees.

Let $\mathit{v}=\left(\begin{array}{c}t\\ -3\\ t+2\end{array}\right)$ and $\mathit{w}=\left(\begin{array}{c}-6\\ 7\\ t\end{array}\right)$.

Given that $\mathit{v}$ and $\mathit{w}$ are perpendicular, find all possible values of $t$.

Show that the angle between $\mathit{v}$ and $\mathit{w}$ is acute for all $t>7$.

Consider the vectors $\mathit{a}=3\mathit{i}-\mathit{j}+4\mathit{k}$ and $\mathit{b}=\left(2+t\right)\mathit{i}-2\mathit{j}+2t\mathit{k}$.

By finding the vector product, determine the value of $t$, given that $\mathit{a}$ and $\mathit{b}$ are parallel.

Consider the vectors $\mathit{a}=-2\mathit{i}-\mathit{j}+3\mathit{k}$ and $\mathit{b}=3\mathit{i}+5\mathit{k}$.

Find a vector of length 7 that is parallel to $\mathit{a}$.

Find a vector that is normal to both $\mathit{a}$ and $\mathit{b}$.

Consider the vectors $\mathit{r}=\left(\begin{array}{c}2\\ 4\\ -1\end{array}\right)$ and $\mathit{t}=\left(\begin{array}{c}-3\\ 5\\ 3\end{array}\right)$.

Show that $3\mathit{r}\times \mathit{t}=3\left(\mathit{r}\times \mathit{t}\right)$.

Find the area of a triangle which has vectors $3\mathit{r}$ and $\mathit{t}$ as two of its sides.

On a calm day, a remote-controlled boat is being driven in the direction of the vector $\mathit{u}=\mathit{i}+3\mathit{j}$ from one side of a pond to the other.

The boat is retrieved and taken to the same starting point, to make the journey again but this time a steady wind causes the boat to travel in a direction represented by the vector $\mathit{w}=2\mathit{i}-\mathit{j}$.

Calculate the angle, in degrees, between the direction of travel on its initial journey and the direction on its subsequent journey.

During the first journey, the boat takes 6.3 seconds to travel the 7.56 m to the other side of the pond. 

Find the velocity vector of the boat.

Given that during the second journey the boat covers a distance of 5.1 m, find the distance between the end points for both journeys.

$\mathrm{ABCD}$ is a parallelogram with vertices $\mathrm{A}\left(2,3,0\right)$, $\mathrm{B}\left(3,9,4\right)$, $\mathrm{C}\left(7,4,2\right)$ and $\mathrm{D}\left(6,-2,-2\right)$.

Find the vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AD}}$.

Find the area of the parallelogram.

By finding the scalar product of $\overrightarrow{\mathrm{BA}}$ and $\overrightarrow{\mathrm{BC}}$, determine if the angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is acute or obtuse.

The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ have position vectors $\mathit{a}$, $\mathit{b}$, $\mathit{c}$ and $\mathit{d}$, relative to the origin $\mathrm{O}$.

The position vectors are given by

$\mathit{a}=2\mathit{i}+4\mathit{j}-\mathit{k}$

$\mathit{b}=-r\mathit{i}+\mathit{j}+2\mathit{k}$

$\mathit{c}=3\mathit{i}+s\mathit{j}$

$\mathit{d}=2\mathit{i}-2\mathit{j}-t\mathit{k}$

where $r$, $s$ and $t$ are constants.

Given that $\overrightarrow{\mathrm{BA}}=\overrightarrow{\mathrm{CD}}$, find $r$, $s$ and $t$.

A fifth point, $\mathrm{E}$, has position vector $\mathit{e}$, relative to the origin $\mathrm{O}$.

Given that $\overrightarrow{\mathrm{AE}}=3\overrightarrow{\mathrm{CD}}$, find the position vector of $\mathrm{E}$.

Find the unit vector that has the same direction as $\mathit{e}$.

A parallelogram $\mathrm{ABCD}$ is shown in the diagram below.

$\overrightarrow{\mathrm{AB}}=\mathit{a}$ and $\overrightarrow{\mathrm{AD}}=\mathit{b}$.

The point $\mathrm{X}$ lies on $\left[\mathrm{BD}\right]$ such that $\mathrm{BX}:\mathrm{XD}=2:1$.

Express $\overrightarrow{\mathrm{CX}}$ in terms of $\mathit{a}$ and $\mathit{b}$.

The point $\mathrm{Y}$ lies on $\left[\mathrm{CD}\right]$ such that $\mathrm{A}$, $\mathrm{X}$ and $\mathrm{Y}$ are collinear.

Express $\overrightarrow{\mathrm{AY}}$ in terms of $\mathit{a}$ and $\mathit{b}$.

Three points have coordinates $\mathrm{A}\left(0,5\right)$, $\mathrm{B}\left(6,4\right)$ and $\mathrm{C}\left(8,16\right)$.

(i) Find the magnitude of $\overrightarrow{\mathrm{AB}}$.

(ii) Find the magnitude of $\overrightarrow{\mathrm{BC}}$.

Given that $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is a right angle, find the area of triangle $\mathrm{ABC}$.

The points $\mathrm{A}$ and $\mathrm{B}$ have coordinates $\left(4,-7,n\right)$ and $\left(-1,3p,2\right)$ respectively.

Given that $\overrightarrow{\mathrm{AB}}=-5\mathit{i}+19\mathit{j}+6\mathit{k}$, find the value of $n$ and the value of $p$.

Consider the unit vector $\mathit{v}=\frac{4}{9}\mathit{i}+q\mathit{j}-\frac{7}{9}\mathit{k}$, where $q>0$.

The point $\mathrm{C}$ is such that $\overrightarrow{\mathrm{AC}}=18\mathit{v}$.

Find the coordinates of $\mathrm{C}$.

Given $\left|\mathit{a}\right|=5$, $\mathit{b}=\left(\begin{array}{c}12\\ -5\end{array}\right)$ and $\mathit{a}·\mathit{b}=16$, find the angle between $\mathit{a}$ and $\mathit{b}$.

Consider a third vector $\mathit{c}$, where $\left|\mathit{c}\right|=8$.

When the angle between $\mathit{a}$ and $\mathit{c}$ is $\frac{\pi }{4}$, show that $\left|\mathit{a}·\left(\mathit{b}+\mathit{c}\right)\right|=16+20\sqrt{2}$.

Two points $\mathrm{A}$ and $\mathrm{B}$ have position vectors $\mathit{a}=\left(\begin{array}{c}6\\ 2\\ -5\end{array}\right)$ and $\mathit{b}=\lambda \left(\begin{array}{c}3\\ -2\\ -1\end{array}\right)$ respectively.

A third point $\mathrm{C}$ is located such that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}8\\ -2\\ 0\end{array}\right)$.

Given that the angle between the vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$ is obtuse, find the range of possible values for $\lambda$.

The points $\mathrm{A}\left(-2,5,3\right)$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ form a parallelogram.

$\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}6\\ 2\\ -1\end{array}\right),\overrightarrow{\mathrm{BC}}=\left(\begin{array}{c}-4\\ 3\\ 4\end{array}\right)$

Find the area of the parallelogram.

Show that the diagonals of the parallelogram are perpendicular to each other.

Determine whether angle $\mathrm{C}\hat{\mathrm{D}}\mathrm{A}$ is acute or obtuse.

Point $\mathrm{A}$ has position vector $\mathit{a}=3\mathit{i}+2\mathit{j}-\mathit{k}$ and point $\mathrm{B}$ has position vector $\mathit{b}=\mathit{i}-\mathit{j}+4\mathit{k}$ relative to the origin $\mathrm{O}$.

Find the area of the triangle $\mathrm{AOB}$.

Point $\mathrm{X}$ is located a distance of 8 units from the origin in the direction perpendicular to the plane formed by $\mathrm{AOB}$.

Find all possible vectors $\overrightarrow{\mathrm{OX}}$.

Find the volume of the tetrahedron $\mathrm{AOBX}$. Give your answer in the form $c\sqrt{d}$, where $c,d\in \mathrm{\mathbb{Z}}$.

The points $\mathrm{A}$ and $\mathrm{B}$ have position vectors $\mathit{a}$ and $\mathit{b}$ respectively.

$\left|\mathit{a}\right|=9,\mathit{a}\times \mathit{b}=\left(\begin{array}{c}5\\ -3\\ 2\end{array}\right),\mathit{a}·\mathit{b}=16$

Find the angle between $\mathit{a}$ and $\mathit{b}$.

The points $\mathrm{A}$ and $\mathrm{B}$ form a triangle with the origin $\mathrm{O}$.

Find $\left|\mathit{b}\right|$.

$\mathrm{ABCD}$ is a parallelogram with vertices $\mathrm{A}\left(-2,5,3\right)$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}\left(3,-1,t\right)$, where $t>0$.

$\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{DC}}=\left(\begin{array}{c}2\\ 4\\ 1\end{array}\right)$

Given that the area of the parallelogram is $\sqrt{1221}$, find the value of $t$.

In vector form, the two parallel sides of a trapezium are given by $\mathit{u}=\left(\begin{array}{c}t\\ 2t-6\\ 1\end{array}\right)$ and $\mathit{v}=\left(\begin{array}{c}t-6\\ 4\\ t-4\end{array}\right)$. Additionally, $\left|\mathit{u}\right|\left|\mathit{v}\right|=18$.

Given that $t$ is an integer, find the value of $t$.

A third side of the trapezium, with vector $\mathit{w}=\left(\begin{array}{c}x\\ 3\\ z\end{array}\right)$, is perpendicular to both $\mathit{u}$ and $\mathit{v}$.

Given that $\left|\mathit{w}\right|=\sqrt{26}$, and that $x$ is an integer, find the values of $x$ and $z$.

$\mathrm{ABCDEFGH}$ is a cuboid as shown in the diagram below.

Point $\mathrm{A}$ is located at $\left(5,-3,4\right)$, $\overrightarrow{\mathrm{AB}}=2\mathit{j}-\mathit{k}$ and $\overrightarrow{\mathrm{BC}}=4\mathit{i}+\mathit{j}+2\mathit{k}$.

The perpendicular distance between the faces $\mathrm{ABCD}$ and $\mathrm{EFGH}$ of the cuboid is $2\sqrt{105}$ units.

Find the coordinates of the point $\mathrm{E}$, given that its $z$-coordinate is positive.

A triangle is formed inside the cuboid by connecting the vertices $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{E}$, where $\mathrm{B}\hat{\mathrm{C}}\mathrm{E}=\theta$.

Using vector methods, find $\cos \theta$.

The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ form the vertices of a parallelogram with position vectors $\mathit{a}$, $\mathit{b}$, $\mathit{c}$ and $\mathit{d}$ respectively.

Show that the area of the parallelogram is $\left|\mathit{a}\times \mathit{b}+\mathit{b}\times \mathit{d}+\mathit{d}\times \mathit{a}\right|$.

Hence show that the shortest distance from $\mathrm{B}$ to $\left(\mathrm{AD}\right)$ is $\frac{\left|\mathit{a}\times \mathit{b}+\mathit{b}\times \mathit{d}+\mathit{d}\times \mathit{a}\right|}{\left|\mathit{d}-\mathit{a}\right|}$.

The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ have position vectors

$\mathit{a}=3\mathit{i}+2\mathit{j}+7\mathit{k}$

$\mathit{b}=\mathit{i}+5\mathit{j}+\mathit{k}$

$\mathit{c}=4\mathit{i}-4\mathit{j}+10\mathit{k}$

$\mathit{d}=2\mathit{j}-2\mathit{k}$

respectively.

(i) Find $\overrightarrow{\mathrm{CD}}$.

(ii) Show that $\overrightarrow{\mathrm{AB}}=\lambda \overrightarrow{\mathrm{CD}}$ and state the value of $\lambda$.

A fifth point $\mathrm{E}\left(p,q,r\right)$ is such that $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{E}$ lie on a straight line and $\mathrm{BE}=21$.

Find the possible values of $p$, $q$ and $r$.

$\mathrm{ABCDEFGH}$ are vertices of a parallelepiped with the vectors $\overrightarrow{\mathrm{AB}}$, $\overrightarrow{\mathrm{AD}}$ and $\overrightarrow{\mathrm{AE}}$ defined as $\mathit{p}$, $\mathit{q}$ and $\mathit{r}$ respectively. $\theta$ is the angle between $\overrightarrow{\mathrm{AE}}$ and the normal to the base $\mathrm{ABCD}$. This information can be seen in the diagram below.

Find an expression for

(i) the area of the base $\mathrm{ABCD}$,

(ii) the perpendicular height of the parallelepiped.

Hence, show that the volume of a parallelepiped is given by $\left|\left(\mathit{p}\times \mathit{q}\right)·\mathit{r}\right|$ units³.

Find the volume of a parallelepiped with vertices $\mathrm{A}\left(5,7,-3\right)$, $\mathrm{B}\left(6,10,-2\right)$, $\mathrm{C}\left(9,11,0\right)$ and $\mathrm{E}\left(4,5,4\right)$.

The points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ have position vectors $\overrightarrow{\mathrm{OA}}=\left(\begin{array}{c}-2\\ 5\end{array}\right)$, $\overrightarrow{\mathrm{OB}}=\left(\begin{array}{c}1\\ 3\end{array}\right)$ and $\overrightarrow{\mathrm{OC}}=\left(\begin{array}{c}4\\ 14\end{array}\right)$ respectively.

Given that triangle $\mathrm{ABC}$ is right-angled, find the area of triangle $\mathrm{ABC}$.

A fourth point, $\mathrm{D}$, is such that $\overrightarrow{\mathrm{BD}}=5\overrightarrow{\mathrm{CA}}$.

Find the coordinates of $\mathrm{D}$.

In the diagram below, $\left[\mathrm{OXA}\right]$, $\left[\mathrm{OYB}\right]$ and $\left[\mathrm{ABC}\right]$ are straight lines.

The position vectors of $\mathrm{A}$ and $\mathrm{B}$ relative to $\mathrm{O}$ are $6\mathit{a}$ and $4\mathit{b}$ respectively.

$\mathrm{Y}$ is the midpoint of $\left[\mathrm{OB}\right]$ and $\overrightarrow{\mathrm{AB}}=2\overrightarrow{\mathrm{BC}}$.

The point $\mathrm{X}$ on $\left[\mathrm{OA}\right]$ is such that $\overrightarrow{\mathrm{XA}}=\frac{1}{k}\overrightarrow{\mathrm{OA}}$.

Given that $\left[\mathrm{XYC}\right]$ is a straight line, find the value of $k$.

In the diagram below, $\overrightarrow{\mathrm{OA}}=\mathit{a}$ and $\overrightarrow{\mathrm{OB}}=\mathit{b}$. $\mathrm{Y}$ is the midpoint of $\left[\mathrm{OA}\right]$. $\mathrm{C}$ is a point on $\left[\mathrm{OB}\right]$ such that $\mathrm{OC}:\mathrm{CB}=3:2$.

Find

(i) $\overrightarrow{\mathrm{OC}}$ in terms of $\mathit{a}$ and $\mathit{b}$

(ii) $\overrightarrow{\mathrm{BY}}$ in terms of $\mathit{a}$ and $\mathit{b}$.

$\mathrm{X}$ is a point on $\left[\mathrm{AC}\right]$. It is given that $\overrightarrow{\mathrm{BX}}=\alpha \overrightarrow{\mathrm{BY}}$ and $\overrightarrow{\mathrm{CX}}=\beta \overrightarrow{\mathrm{CA}}$, where $\alpha ,\beta \in \mathrm{ℝ}$.

Find an expression for $\overrightarrow{\mathrm{OX}}$

(i) in terms of $\mathit{a}$, $\mathit{b}$ and $\alpha$

(ii) in terms of $\mathit{a}$, $\mathit{b}$ and $\beta$.

Hence find the values of $\alpha$ and $\beta$.

Hence find an expression for $\overrightarrow{\mathrm{XY}}$ in terms of $\mathit{a}$ and $\mathit{b}$.

A parallelogram $\mathrm{ABCD}$ has vertices $\mathrm{A}\left(-1,7\right)$, $\mathrm{B}\left(2,12\right)$, $\mathrm{C}$ and $\mathrm{D}$, where $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}12\\ 7\end{array}\right)$.

Find the coordinates of $\mathrm{C}$ and the coordinates of $\mathrm{D}$.

Find the perpendicular distance from $\mathrm{B}$ to $\left(\mathrm{AD}\right)$.

Hence find the area of parallelogram $\mathrm{ABCD}$.

Given $\left|\mathit{a}\right|=6$, $\left|\mathit{b}\right|=2$ and $\mathit{a}\times \mathit{b}=\left(\begin{array}{c}-5\\ 2\\ 7\end{array}\right)$, find the possible values of $\mathit{a}·\mathit{b}$.

Consider a third vector $\mathit{c}$.

Given that $\mathit{c}=2\mathit{a}+\mathit{b}$, find $\left|\mathit{a}\times \left(\mathit{b}+\mathit{c}\right)\right|$.

The points $\mathrm{A}$ and $\mathrm{B}$ have position vectors $\mathit{a}$ and $\mathit{b}$ respectively.

$\left|\mathit{a}\right|=\sqrt{35}$, $\mathit{b}=2\mathit{i}+2\mathit{j}+\mathit{k}$, $\mathit{a}\times \mathit{b}=\left(\begin{array}{c}7\\ -11\\ 8\end{array}\right)$ and $\theta$ is the acute angle between $\mathit{a}$ and $\mathit{b}$.

Find $\cos \theta$.

A third point $\mathrm{C}$ is located such that its position vector is $\mathit{c}=\left(\begin{array}{c}28\\ -44\\ 32\end{array}\right)$.

Show that $\mathit{c}=4\mathit{a}\times \mathit{b}$.

Show that for any two vectors $\mathit{a}$ and $\mathit{b}$, ${\left(\mathit{a}·\mathit{b}\right)}^2-{\left|\mathit{a}\times \mathit{b}\right|}^2={\left|\mathit{a}\right|}^2{\left|\mathit{b}\right|}^2\cos 2\theta$, where $\theta$ is the angle between $\mathit{a}$ and $\mathit{b}$.

Hence show that when $\theta =\frac{\pi }{4}$, $\left|\mathit{a}\times \mathit{b}\right|=\mathit{a}·\mathit{b}$.

Consider a parallelepiped $\mathrm{ABCDEFGH}$ with vertices $\mathrm{A}\left(2,5,-1\right)$, $\mathrm{B}\left(3,8,0\right)$, $\mathrm{C}\left(5,4,3\right)$ and $\mathrm{H}\left(5,2,7\right)$ as seen in the diagram below.

By first finding the perpendicular height of the parallelepiped, find its volume.

$\mathrm{ABCDE}$ is a pentagon, where $\mathrm{A}\left(0,4,-1\right)$, $\mathrm{D}\left(-1,7,3\right)$, $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}-4\\ 2\\ 5\end{array}\right)$ and $\overrightarrow{\mathrm{CD}}=\left(\begin{array}{c}1\\ -1\\ -1\end{array}\right)$.

Given that $\overrightarrow{\mathrm{BD}}=\overrightarrow{\mathrm{AE}}$, find the area of triangle $\mathrm{BCD}$ as a percentage of the total area of the pentagon.

Three points, $\mathrm{A}\left(-3,-1\right)$, $\mathrm{B}\left(2,2\right)$ and $\mathrm{C}$, are located on a straight line where $\overrightarrow{\mathrm{AC}}=\lambda \overrightarrow{\mathrm{AB}}$ and $\lambda >1$. A fourth point $\mathrm{D}$ lies in the fourth quadrant and is located such that $\overrightarrow{\mathrm{DB}}$ is perpendicular to $\overrightarrow{\mathrm{OB}}$ and $\mathrm{DB}=2\sqrt{22}$.

Find $\mathrm{D}\hat{\mathrm{B}}\mathrm{C}$.

Given that the area of triangle $\mathrm{BCD}$ is $16\sqrt{11}$, find $\lambda$.

Consider a regular hexagon $\mathrm{ABCDEF}$ with sides of length $9\sqrt{2}$ units. The position vectors of $\mathrm{A}$ and $\mathrm{E}$ are $\mathit{a}=-11\mathit{i}-4\mathit{j}+5\mathit{k}$ and $\mathit{e}=3\mathit{i}-3\mathit{j}-12\mathit{k}$ respectively.

Given that the coordinates of $\mathrm{F}$ are $\left(r,t,t\right)$, where $r,t\in \mathrm{\mathbb{Z}}$ and $r\ne 0$, find the values of $r$ and $t$.

$\mathrm{ABCD}$ is a parallelogram defined by the vectors $\overrightarrow{\mathrm{AB}}=\mathit{p}$ and $\overrightarrow{\mathrm{AD}}=\mathit{q}$, where $\mathit{p}=\left(\begin{array}{c}1\\ 5\end{array}\right)$ and $\mathit{q}=\left(\begin{array}{c}2t+2\\ t\end{array}\right)$.

Given that the angle $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}$ is acute, find the range of values for $t$.

$\overrightarrow{\mathrm{AB}}$ is enlarged by a factor of $k$.

Show that $k\mathit{p}·\mathit{q}=k\left(\mathit{p}·\mathit{q}\right)$.

Given that $k=7$, find the range of possible values for the area of the enlarged parallelogram.

Consider the parallelepiped $\mathrm{ABCDEFGH}$ as shown in the diagram below. The position vectors of $\mathrm{A}$, $\mathrm{C}$, $\mathrm{D}$ and $\mathrm{E}$ are $\mathit{a}=-5\mathit{i}+2\mathit{j}-4\mathit{k}$, $\mathit{c}=2\mathit{i}+3\mathit{j}+\mathit{k}$, $\mathit{d}=-\mathit{i}+\mathit{j}$ and $\mathit{e}=-3\mathit{i}+\mathit{j}+2\mathit{k}$ respectively.

$\mathrm{X}$ is a point located on the line $\left[\mathrm{EC}\right]$ such that $\overrightarrow{\mathrm{EX}}=\lambda \overrightarrow{\mathrm{EC}}$.

Find the shortest distance from $\mathrm{H}$ to $\left[\mathrm{EC}\right]$.

Consider a tetrahedron $\mathrm{ABCV}$ where $\overrightarrow{\mathrm{AB}}=\mathit{p}$, $\overrightarrow{\mathrm{AC}}=\mathit{q}$ and $\overrightarrow{\mathrm{AV}}=\mathit{r}$. The perpendicular height, $h$, of the tetrahedron from the base $\mathrm{ABC}$ makes an angle of $\theta$ with $\mathit{r}$.

This information is shown in the diagram below.

Find an expression for the volume of the tetrahedron in terms of $\mathit{p}$, $\mathit{q}$ and $\mathit{r}$.

Find the volume of the tetrahedron when $\mathit{p}=\left(\begin{array}{c}4\\ 5\\ -2\end{array}\right)$, $\mathit{q}=\left(\begin{array}{c}2\\ 2\\ -5\end{array}\right)$ and $\mathit{r}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$.

Hence find the shortest distance between vertex A and its opposite face.