Vectors in 3D (AQA A Level Maths: Pure): Exam Questions

Figure 1 shows a sketch of a cube with vertices $O$, $A$, $B$, $C$, $D$, $E$, $F$ and $G$.

Relative to a fixed origin $O$, the edges $OA$, $OB$ and $OC$ represent the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. The vertex $E$ is diagonally opposite $O$, and the vertex $G$ is diagonally opposite $A$.

Find the vectors $\overrightarrow{OE}$ and $\overrightarrow{AG}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.

Let $P$ be a point on the line segment $OE$ and let $Q$ be a point on the line segment $AG$.

Explain why the position vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ can be expressed in the forms

$\overrightarrow{OP}=\lambda \overrightarrow{OE}$

$\overrightarrow{OQ}=\mathbf{a}+\mu \overrightarrow{AG}$

where $\lambda$ and $\mu$ are scalar constants such that $0\le \lambda \le 1$ and $0\le \mu \le 1$.

By solving the equation $\overrightarrow{OP}=\overrightarrow{OQ}$, using your results from (a) and (b), show that the diagonals $OE$ and $AG$ intersect each other, and determine the ratio into which they are cut by their point of intersection.

Relative to a fixed origin $O$, the points $A$ and $B$ have coordinates $\left(-1,3,14\right)$ and $\left(2k,3k,13\right)$ respectively, where $k$ is an integer.

Given that the distance $AB$ is $\sqrt{163}$, find the value of $k$.

Relative to a fixed origin $O$, the points $A$ and $B$ have position vectors $(3\mathbf{i}-2\mathbf{j}+\mathbf{k})$ and $(5\mathbf{i}+\mathbf{j}-5\mathbf{k})$ respectively.

Find the magnitude of the vector $AB$.

The point $C$ has position vector $(a\mathbf{i}+b\mathbf{j}+2\mathbf{k})$. Given that $A$, $B$ and $C$ are collinear, find the values of the constants $a$ and $b$.

Relative to a fixed origin $O$, the points $A$ and $B$ have coordinates $\left(-2,5,-7\right)$ and $\left(-7,k,3\right)$ respectively, where $k$ is a constant.

Given that the distance $AB$ is $5\sqrt{14}$, find the possible values of $k$.

Figure 2 shows a sketch of triangle $ABC$.

Given that

$\overrightarrow{AB}=-2\mathbf{i}+3\mathbf{j}-\mathbf{k}$

$\overrightarrow{AC}=-5\mathbf{i}-4\mathbf{j}-7\mathbf{k}$

show that the size of angle $BAC$ is $81.9°$ to one decimal place.

Relative to a fixed origin $O$, the point $R$ has coordinates $\left(-1,5,14\right)$ and the point $S$ has coordinates $\left(7,-2,12\right)$.

Find:

(i) the vector $\overrightarrow{RS}$,

(ii) a unit vector in the direction of $\overrightarrow{RS}$.

Find the angle that $\overrightarrow{RS}$ makes with the positive $y$-axis. Give your answer in degrees to one decimal place.

The vector $\overrightarrow{TU}$ is given by

$\overrightarrow{TU}=-24\mathbf{i}+21\mathbf{j}+6\mathbf{k}$

Explain, giving a reason for your answer, whether the vectors $\overrightarrow{RS}$ and $\overrightarrow{TU}$ are parallel.

Relative to a fixed origin $O$, the points $P$, $Q$ and $R$ have coordinates $\left(12,3,-3\right)$, $\left(7,-8,k\right)$ and $\left(3,3,-12\right)$ respectively, where $k$ is a constant.

Given that triangle $PQR$ is an equilateral triangle, find the value of $k$.

The vectors $\mathbf{a}$ and $\mathbf{b}$ are defined by

$\mathbf{a}=-12\mathbf{i}-7\mathbf{j}+15\mathbf{k}$

$\mathbf{b}=4p\mathbf{i}+(pqr+2qr-p)\mathbf{j}-pq\mathbf{k}$

where $p$, $q$ and $r$ are scalar constants.

Given that $\mathbf{a}=\mathbf{b}$, find the value of $p$, the value of $q$ and the value of $r$.

A particle $P$ of mass $0.4\text{ kg}$ is acted upon by a force ${\mathbf{F}}_1$ where

${\mathbf{F}}_1=\left(-2\mathbf{i}+6\mathbf{j}+10\mathbf{k}\right)\text{ N}$

Find:

(i) the acceleration of $P$ while the force acts,

(ii) the magnitude of the acceleration of $P$, giving your answer to 3 significant figures.

A second force ${\mathbf{F}}_2$ with a magnitude of $10\sqrt{3}\text{ N}$ now acts on $P$. The resultant of ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ is parallel to the vector $\mathbf{j}$ and has a magnitude of $8\text{ N}$.

Find ${\mathbf{F}}_2$, giving your answer in the form $\left(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\right)\text{ N}$.

Three forces, ${\mathbf{F}}_1$, ${\mathbf{F}}_2$ and ${\mathbf{F}}_3$, act on a particle of mass $5\text{ kg}$. The forces are given by

${\mathbf{F}}_1=\left(3\mathbf{i}-7\mathbf{j}+p\mathbf{k}\right)\text{ N}$

${\mathbf{F}}_2=\left(q\mathbf{i}+3\mathbf{j}-\mathbf{k}\right)\text{ N}$

${\mathbf{F}}_3=\left(-2\mathbf{i}+r\mathbf{j}-5\mathbf{k}\right)\text{ N}$

where $p$, $q$ and $r$ are scalar constants.

Under the action of these three forces, the particle is in equilibrium.

Find the value of $p$, the value of $q$ and the value of $r$.

The third force is now doubled, so that the three forces acting on the particle are ${\mathbf{F}}_1$, ${\mathbf{F}}_2$ and $2{\mathbf{F}}_3$.

Find:

(i) the resultant force $\mathbf{R}$ now acting on the particle,

(ii) the acceleration of the particle,

(iii) the magnitude of the acceleration of the particle, giving your answer as an exact value.

Figure 1 shows a sketch of triangle $ABC$.

Given that

$\overrightarrow{AB}=7\mathbf{i}+\mathbf{j}-\mathbf{k}$

$\overrightarrow{AC}=-2\mathbf{i}+5\mathbf{k}$

show that the size of angle $BAC$ is $119.6°$ to one decimal place.

Hence find the area of triangle $ABC$, giving your answer to 3 significant figures.

Relative to a fixed origin $O$, the points $A$, $B$ and $C$ have position vectors

$\mathbf{a}=2\mathbf{i}-\mathbf{j}+3\mathbf{k}$

$\mathbf{b}=4\mathbf{i}+2\mathbf{j}-\mathbf{k}$

$\mathbf{c}=3\mathbf{i}+5\mathbf{k}$

Find the vector $AB$.

The point $D$ is such that $ABCD$ is a parallelogram.

Find the position vector of $D$.

Find the size of angle $ABC$, giving your answer in degrees to 1 decimal place.

Relative to a fixed origin $O$, the point $R$ has position vector $\mathbf{i}+6\mathbf{j}-2\mathbf{k}$ and the point $S$ has position vector $10\mathbf{i}+13\mathbf{k}$.

Find a unit vector in the direction of $\overrightarrow{RS}$.

Find the angle that $\overrightarrow{RS}$ makes with the negative $z$-axis. Give your answer in degrees to one decimal place.

The vector $\overrightarrow{TU}$ is given by

$\overrightarrow{TU}=-12\mathbf{i}+8\mathbf{j}-20\mathbf{k}$

Explain, giving a reason for your answer, whether the vectors $\overrightarrow{RS}$ and $\overrightarrow{TU}$ are parallel.

Relative to a fixed origin $O$, the points $P$, $Q$ and $R$ have coordinates $\left(2,-3,1\right)$, $\left(-1,-4,3\right)$ and $\left(k,0,3\right)$ respectively, where $k$ is a constant.

Given that triangle $PQR$ is isosceles, and that $k>1$, find the value of $k$.

The vectors $\mathbf{a}$ and $\mathbf{b}$ are defined by

$\mathbf{a}=(p+1)\mathbf{i}-7\mathbf{j}+(q-3p)\mathbf{k}$

$\mathbf{b}=5\mathbf{i}+(14q+r)\mathbf{j}+(1-2r)\mathbf{k}$

where $p$, $q$ and $r$ are scalar constants.

Given that $\mathbf{a}=\mathbf{b}$, find the value of $p$, the value of $q$ and the value of $r$.

A particle $P$ of mass $0.5\text{ kg}$ is acted upon by a force ${\mathbf{F}}_1$ where

${\mathbf{F}}_1=\left(12\mathbf{i}-4\mathbf{j}+p\mathbf{k}\right)\text{ N}$

and $p$ is a scalar constant.

Given that the magnitude of the acceleration of $P$ is $26{\text{ m s}}^{-2}$, find the possible values of $p$.

An additional second force ${\mathbf{F}}_2=q\mathbf{k}\text{ N}$, where $q$ is a constant and $q<0$, now acts on $P$. Under the action of the resultant of these two forces, $P$ now experiences an acceleration of magnitude $8\sqrt{10}{\text{ m s}}^{-2}$.

Explain why this additional information shows that $p>0$.

Hence find the value of $q$.

Three forces, ${\mathbf{F}}_1$, ${\mathbf{F}}_2$ and ${\mathbf{F}}_3$, act on a particle of mass $5\text{ kg}$. The forces are given by

${\mathbf{F}}_1=\left(r\mathbf{i}+5\mathbf{j}+(r-p)\mathbf{k}\right)\text{ N}$

${\mathbf{F}}_2=\left((p+q)\mathbf{i}-3p\mathbf{j}+7\mathbf{k}\right)\text{ N}$

${\mathbf{F}}_3=\left(-\mathbf{i}+r\mathbf{j}-2q\mathbf{k}\right)\text{ N}$

where $p$, $q$ and $r$ are scalar constants.

Under the action of these three forces, the particle is in equilibrium.

Find the value of $p$, the value of $q$ and the value of $r$.

A fourth force, ${\mathbf{F}}_4$, is now added to the particle. Under the combined action of the four forces, the particle experiences an acceleration with a magnitude of $2.2{\text{ m s}}^{-2}$ in the same direction as the vector $\mathbf{i}-\mathbf{j}+3\mathbf{k}$.

Find ${\mathbf{F}}_4$, giving your answer in the form $\left(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\right)\text{ N}$, where $x$, $y$ and $z$ are exact values.

Figure 2 shows a sketch of a cube with vertices $O$, $A$, $B$, $C$, $D$, $E$, $F$ and $G$.

Relative to a fixed origin $O$, the position vectors of the vertices $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. The face of the cube containing $OA$ and $OB$ is $OADB$. The face of the cube containing $OA$ and $OC$ is $OAFC$.

Using vector methods, prove that the diagonals $CD$ and $BF$ bisect each other.

Relative to a fixed origin $O$, the points $A$ and $B$ have coordinates $\left(5,0,-1\right)$ and $\left(k,-2,3k\right)$ respectively, where $k$ is a constant.

Given that the distance $AB$ is $6\sqrt{k}$, and that the point $B$ lies at a distance of $\sqrt{14}$ from $O$, find the value of $k$.

The vectors $\mathbf{a}$ and $\mathbf{b}$ are defined by

$\mathbf{a}=pq\mathbf{i}-24\mathbf{j}+9r\mathbf{k}$

$\mathbf{b}=6\mathbf{i}+(p-r)\mathbf{j}+(p-3q)\mathbf{k}$

where $p$, $q$ and $r$ are scalar constants.

Given that $\mathbf{a}=3\mathbf{b}$, and that $r>0$, find the value of $p$, the value of $q$ and the value of $r$.

Figure 1 shows a sketch of a parallelogram $ABCD$.

Given that

$\overrightarrow{AB}=3\mathbf{i}-\mathbf{j}-2\mathbf{k}$

$\overrightarrow{AD}=7\mathbf{i}-\mathbf{j}+4\mathbf{k}$

find the area of the parallelogram $ABCD$. Give your answer to 3 significant figures.

The vector $\overrightarrow{RS}$ is given by

$\overrightarrow{RS}=x\mathbf{i}-9\mathbf{j}+3\mathbf{k}$

where $x$ is a constant.

The vector $\overrightarrow{RS}$ makes an angle $\theta$ with the positive $x$-axis.

Show that $x^2=\frac{90}{{\text{tan}}^2\theta }$.

Given further that $\theta$ is acute and that $\text{cos}\theta =\frac{4}{5}$, find a unit vector in the direction of $\overrightarrow{RS}$.

Figure 2 shows a sketch of a regular tetrahedron.

Relative to a fixed origin $O$, the points $A$, $B$, $C$ and $D$ have coordinates $\left(1,1,1\right)$, $\left(-8,10,1\right)$, $\left(-3,6,-10\right)$ and $\left(-2k,13,k\right)$ respectively, where $k$ is a constant.

Find the coordinates of the point $D$.

A particle $P$ of mass $0.2\text{ kg}$ is acted upon by a force ${\mathbf{F}}_1$ where

${\mathbf{F}}_1=\left(-3\mathbf{i}+p\mathbf{j}+4\mathbf{k}\right)\text{ N}$

and $p$ is a scalar constant.

Given that the magnitude of the acceleration of $P$ is $65{\text{ m s}}^{-2}$, find the possible values of $p$.

An additional second force ${\mathbf{F}}_2=q\mathbf{j}\text{ N}$, where $q$ is a constant and $q>0$, now acts on $P$. Under the action of the resultant of these two forces, $P$ experiences an acceleration of magnitude $\frac{5\sqrt{221}}{2}{\text{ m s}}^{-2}$.

Find the possible values of $q$.

[In this question, the unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively, and $\mathbf{k}$ is a unit vector directed vertically upwards.]

A robotic submarine of mass $750\text{ kg}$ is initially moving in a level direction such that the $\mathbf{k}$ component of its velocity is zero.

In addition to its weight, the forces acting on the submarine are the combined thrust and lift $\mathbf{T}$, its buoyancy $\mathbf{B}$, and the water resistance $\mathbf{W}$. These forces, measured in newtons, are given by

$\mathbf{T}=600\mathbf{i}-750\mathbf{j}-120\mathbf{k}$

$\mathbf{B}=7360\mathbf{k}$

$\mathbf{W}=-500\mathbf{i}+600\mathbf{j}+50\mathbf{k}$

Taking $g=9.8{\text{ m s}}^{-2}$, find the magnitude of the acceleration of the submarine. Give your answer to 3 significant figures.

Determine whether the submarine is rising or sinking, giving a reason for your answer, and find the angle its acceleration makes with the vector $\mathbf{k}$. Give your angle in degrees to one decimal place.

Figure 3 shows a sketch of a cuboid with vertices $O$, $A$, $B$, $C$, $D$, $E$, $F$ and $G$.

Relative to a fixed origin $O$, the position vectors of the vertices $A$, $B$ and $C$ are $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively. The face of the cuboid containing $OA$ and $OC$ is $OAFC$. The vertex $E$ is diagonally opposite $O$.

The point $P$ lies on the space diagonal $OE$ such that it divides $OE$ in the ratio $a:b$, where $a$ and $b$ are positive constants with $a>b$.

Using vector methods, show that if the line segment $BP$ is extended, it will intersect the edge $FE$, and show that $FE$ is divided in the ratio $(a-b):b$ by the point of intersection.