Vectors

The vector equation of line $l$ is $\mathbf{r} = \begin{pmatrix}1\\2\\-1\end{pmatrix} + \lambda \begin{pmatrix}2\\1\\3\end{pmatrix}$. The plane $p$ is given by $\mathbf{r} \cdot \begin{pmatrix}2\\-1\\-1\end{pmatrix} = 6$.

Feb/March 2016

The line $l$ is described by $\mathbf{r} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + \mathbf{k})$. The plane $p$ is described by $3x + y - 5z = 20$.

Feb/March 2017

Line $l$ is described by $\mathbf{r} = 4\mathbf{i} + 3\mathbf{j} - \mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$. Plane $p$ is given by $2x - 3y - z = 4$.

Feb/March 2018

The two planes are given by the equations $2x + 3y - z = 1$ and $x - 2y + z = 3$.

Feb/March 2019

In the diagram, $OABCDEFG$ is a cuboid with $OA = 2$ units, $OC = 3$ units and $OD = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$, respectively. Point $M$ lies on $AB$ so that $MB = 2AM$. $N$ is the midpoint of $FG$.

Feb/March 2020

The two lines are represented by the equations $\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$.

Feb/March 2021

The position vectors of $A$ and $B$ are $2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ respectively. The line $l$ is given by the vector equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \mu(\mathbf{i} - 3\mathbf{j} - 2\mathbf{k})$.

Feb/March 2022

With origin $O$ as the reference point, the points $A$, $B$, $C$ and $D$ are described by the position vectors $\overrightarrow{OA} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}$, $\overrightarrow{OC} = \begin{pmatrix} 1 \\ -2 \\ 5 \end{pmatrix}$ and $\overrightarrow{OD} = \begin{pmatrix} 5 \\ -6 \\ 11 \end{pmatrix}$.

Feb/March 2023

Taking the origin $O$ as the reference point, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = 5\mathbf{i} - 2\mathbf{j} + \mathbf{k}$, $\overrightarrow{OB} = 8\mathbf{i} + 2\mathbf{j} - 6\mathbf{k}$ and $\overrightarrow{OC} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k}$.

Feb/March 2024

The equations of two lines are $\mathbf{r} = \begin{pmatrix} -1 \\ 3 \\ -4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix}$.

Feb/March 2025

The vector equations of the lines $l$ and $m$ are $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + s(\mathbf{i} - \mathbf{j} + 2\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 6\mathbf{j} + \mathbf{k} + t(2\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ respectively.

May/June 2010

Plane $p$ is given by $3x + 2y + 4z = 13$. A further plane $q$, with equation $ax + y + z = 4$, is perpendicular to $p$, where $a$ is a constant.

May/June 2010

The line $l$ is defined by $\mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$. The plane $p$ has equation $3x - y + 2z = 9$. The point at which $l$ meets the plane $p$ is $A$.

May/June 2010

The coordinates of points $A$ and $B$ are $(-1, 2, 5)$ and $(2, -2, 11)$, respectively. The plane $p$ passes through $B$ and is perpendicular to $AB$.

May/June 2011

The two planes are described by the equations $x + 2y - 2z = 7$ and $2x + y + 3z = 5$.

May/June 2011

Taking O as the origin, the vector equations of lines $l$ and $m$ are $\mathbf{r} = 2\mathbf{i} + \mathbf{k} + \lambda(\mathbf{i} - \mathbf{j} + 2\mathbf{k})$ and $\mathbf{r} = 2\mathbf{j} + 6\mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$ respectively.

May/June 2011

Point $P$ is at coordinates $(-1, 4, 11)$, while line $l$ is given by $\mathbf{r} = \begin{pmatrix}1 \\ 3 \\ -4\end{pmatrix} + \lambda \begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix}$.

May/June 2012

The plane equations for $m$ and $n$ are $x + 2y - 2z = 1$ and $2x - 2y + z = 7$ respectively, and the line $l$ is given by $\mathbf{r} = \mathbf{i} + \mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

May/June 2012

The equations of the lines $l$ and $m$ are $\mathbf{r} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} + \mu(a\mathbf{i} + b\mathbf{j} - \mathbf{k})$, respectively, with $a$ and $b$ as constants.

May/June 2012

The points $P$ and $Q$ are specified by position vectors, measured from the origin $O$, as $\overrightarrow{OP} = 7\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}$ and $\overrightarrow{OQ} = -5\mathbf{i} + \mathbf{j} + \mathbf{k}$. Point $A$ is the midpoint of $PQ$. Plane $\Pi$ is perpendicular to the line $PQ$ and goes through $A$.

May/June 2013

Points $A$ and $B$ are represented by the position vectors $2\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ respectively, and plane $p$ is given by $x + y = 5$.

May/June 2013

The diagram illustrates the curve given by $x^3 + xy^2 + ay^2 - 3ax^2 = 0$, where $a$ is a positive constant. The highest point on the curve is $M$.

May/June 2013

The line $l$ is given by $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} + \mathbf{k})$, with $a$ constant. The plane $p$ is defined by $x + 2y + 2z = 6$. Determine the value or values of $a$ for each of the situations below.

May/June 2013

The line $l$ is represented by ${\bf r} = 4\mathbf{i} - \mathbf{j} + 2\mathbf{k} + \lambda(2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k})$. The plane $p$ goes through the point $(4, -1, 2)$ and is at right angles to $l$.

May/June 2014

Using the origin $O$ as the reference point, the position vectors of the points $A$, $B$ and $C$ are $ \overrightarrow{OA} = i + 2j + 3k$, $\overrightarrow{OB} = 2i + 4j + k$ and $\overrightarrow{OC} = 3i + 5j - 3k$.

May/June 2014

The line $l$ is defined by ${\bf r} = {\bf i} + 2{\bf j} - {\bf k} + \lambda(3{\bf i} - 2{\bf j} + 2{\bf k})$ and the plane $p$ is given by $2x + 3y - 5z = 18$.

May/June 2014

The straight line $l_1$ goes through the points $(0, 1, 5)$ and $(2, -2, 1)$. The straight line $l_2$ is given by $\mathbf{r} = 7\mathbf{i} + \mathbf{j} + k + \mu(\mathbf{i} + 2\mathbf{j} + 5\mathbf{k})$.

May/June 2015

The points $A$ and $B$ are given by the position vectors $\overrightarrow{OA} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OB} = \mathbf{i} + \mathbf{j} + 5\mathbf{k}$. The line $l$ is described by $\mathbf{r} = \mathbf{i} + \mathbf{j} + 2\mathbf{k} + \mu(3\mathbf{i} + \mathbf{j} - \mathbf{k})$.

May/June 2015

The graph of $y = \frac{e^{2x}}{4 + e^{3x}}$ includes a single stationary point.

May/June 2015

The equations of two planes are $x + 3y - 2z = 4$ and $2x + y + 3z = 5$. Their intersection is the straight line $l$.

May/June 2015

Relative to origin $O$, the position vectors of $A$, $B$, $C$, $D$ are $\overrightarrow{OA} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$, $\overrightarrow{OB} = 2\mathbf{i} + \mathbf{j} - \mathbf{k}$, $\overrightarrow{OC} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k}$, $\overrightarrow{OD} = -3\mathbf{i} + \mathbf{j} + 2\mathbf{k}$.

May/June 2016

The curve given by $y = \frac{(\ln x)^2}{x}$ has two stationary points.

May/June 2016

Points $A$, $B$ and $C$ have position vectors relative to the origin $O$, namely $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $\overrightarrow{OB} = 4\mathbf{j} + \mathbf{k}$ and $\overrightarrow{OC} = 2\mathbf{i} + 5\mathbf{j} - \mathbf{k}$. A further point $D$ is chosen so that quadrilateral $ABCD$ is a parallelogram.

May/June 2016

The position vectors of points $A$ and $B$, measured from the origin $O$, are $\vec{OA} = \mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\vec{OB} = 2\mathbf{i} + 3\mathbf{k}$. The line $l$ is given by $\mathbf{r} = 2\mathbf{i} - 2\mathbf{j} - \mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})$.

May/June 2016

The plane given by the equation $2x + 2y - z = 5$ is labelled $m$. Relative to the origin $O$, the coordinates of $A$ and $B$ are $(3, 4, 0)$ and $(-1, 0, 2)$ respectively.

May/June 2017

Measured from the origin $O$, point $A$ has position vector $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$. Line $l$ is defined by $\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})$.

May/June 2017

Points $A$ and $B$ are defined by the position vectors $\overrightarrow{OA} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ and $\overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}$. The line $l$ is described by $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + m\mathbf{k} + \mu(\mathbf{i} - 2\mathbf{j} - 4\mathbf{k})$, where $m$ is a constant.

May/June 2017

Point $P$ is given by the position vector $3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$. Line $l$ is defined by $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + u(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$.

May/June 2018

The lines $l$ and $m$ are given by $\mathbf{r} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} + s(2\mathbf{i} + 3\mathbf{j} - \mathbf{k})$ and $\mathbf{r} = \mathbf{i} + 3\mathbf{j} + 4\mathbf{k} + t(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$, respectively.

May/June 2018

The position vectors of points $A$ and $B$ are $2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ and $4\mathbf{i} + \mathbf{j} + \mathbf{k}$, respectively. The line $l$ is given by $\mathbf{r} = 4\mathbf{i} + 6\mathbf{j} + u(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$.

May/June 2018

The diagram displays rectangular axes $Ox$, $Oy$ and $Oz$, together with four points $A$, $B$, $C$ and $D$, whose position vectors are $\overrightarrow{OA} = 3\mathbf{i}$, $\overrightarrow{OB} = 3\mathbf{i} + 4\mathbf{j}$, $\overrightarrow{OC} = \mathbf{i} + 3\mathbf{j}$ and $\overrightarrow{OD} = 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$.

May/June 2019

The points $A$ and $B$ are given by the position vectors $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $3\mathbf{i} + \mathbf{j} + \mathbf{k}$, respectively. The line $l$ is defined by $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

May/June 2019

The equation of the line $l$ is $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + u(2\mathbf{i} - \mathbf{j} - 2\mathbf{k})$.

May/June 2019

Relative to the origin $O$, triangle $ABC$ has vertices with position vectors $\overrightarrow{OA} = 2i + 5k$, $\overrightarrow{OB} = 3i + 2j + 3k$ and $\overrightarrow{OC} = i + j + k$.

May/June 2020

Using $O$ as the origin, the position vectors of $A$ and $B$ are $ \overrightarrow{OA} = 6\mathbf{i} + 2\mathbf{j}$ and $\overrightarrow{OB} = 2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$. $M$ is the midpoint of $OA$. Point $N$, which lies on $AB$ between $A$ and $B$, satisfies $AN = 2NB$.

May/June 2020

With respect to the origin $O$, the position vectors of $A$, $B$ and $D$ are $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}$, $\overrightarrow{OB} = 2\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OD} = 3\mathbf{i} + 2\mathbf{k}$. A further point $C$ is arranged so that $ABCD$ forms a parallelogram.

May/June 2020

Taking $O$ as the origin, the position vectors of $A$ and $B$ are $\overrightarrow{OA} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$. The line $l$ is given by $\mathbf{r} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$.

May/June 2021

Relative to origin $O$, the position vectors of points $A$ and $B$ are $\overrightarrow{OA} = 2\mathbf{i} - \mathbf{j}$ and $\overrightarrow{OB} = \mathbf{j} - 2\mathbf{k}$.

May/June 2021

The quadrilateral $ABCD$ is a trapezium with $AB \parallel DC$. Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $\overrightarrow{OB} = \mathbf{i} + 3\mathbf{j} + \mathbf{k}$ and $\overrightarrow{OC} = 2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$.

May/June 2021

In the diagram, $OABCDEFG$ forms a cuboid, with $OA = 2$ units, $OC = 4$ units and $OG = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OG$ respectively. Point $M$ is the midpoint of $DF$. Point $N$ lies on $AB$ so that $AN = 3NB$.

May/June 2022

The vector equations for the lines $l$ and $m$ are $\mathbf{r} = -\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 5\mathbf{i} + 4\mathbf{j} + 3\mathbf{k} + \mu(a\mathbf{i} + b\mathbf{j} + \mathbf{k})$ respectively, where $a$ and $b$ are constants.

May/June 2022

With respect to origin $O$, point $A$ has position vector $overrightarrow{OA} = \mathbf{i} + 5\mathbf{j} + 6\mathbf{k}$. The line $l$ is represented by $\mathbf{r} = 4\mathbf{i} + \mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$.

May/June 2022

Measured from origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\vec{OA} = \begin{pmatrix}2\\1\\3\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}4\\3\\2\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}3\\-2\\-4\end{pmatrix}$. The quadrilateral $ABCD$ is a parallelogram.

May/June 2023

The position vectors of points $A$ and $B$ are $\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$ and $2\mathbf{i} - \mathbf{j} + \mathbf{k}$ respectively. The equation of the line $l$ is $\mathbf{r} = \mathbf{i} - \mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})$.

May/June 2023

The equations of the lines $l$ and $m$ are $l: \; \mathbf{r} = a\mathbf{i} + 3\mathbf{j} + b\mathbf{k} + \lambda(c\mathbf{i} - 2\mathbf{j} + 4\mathbf{k})$, $m: \; \mathbf{r} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + \mathbf{k})$. Relative to origin $O$, the position vector of $P$ is $4\mathbf{i} + 7\mathbf{j} - 2\mathbf{k}$.

May/June 2023

Two straight lines, $l_1$ and $l_2$, have the equations $l_1: \mathbf{r} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + a\mathbf{k})$ and $l_2: \mathbf{r} = -\mathbf{i} - \mathbf{j} - \mathbf{k} + \mu(3\mathbf{i} - 2\mathbf{j} - 2\mathbf{k})$, where $a$ is constant. These lines, $l_1$ and $l_2$, are perpendicular.

May/June 2024

Points $A$, $B$ and $C$ are given by position vectors $\overrightarrow{OA} = -2\mathbf{i} + \mathbf{j} + 4\mathbf{k}$, $\overrightarrow{OB} = 5\mathbf{i} + 2\mathbf{j}$ and $\overrightarrow{OC} = 8\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}$, with $O$ as the origin. The line $l_1$ goes through $B$ and $C$.

May/June 2024

The equations for two straight lines are $\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k})$ and $\mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$, with $a$ as a constant.

May/June 2024

With reference to the origin $O$, the position vectors of $A$ and $B$ are $2\mathbf{i} + 4\mathbf{k}$ and $5\mathbf{i} + \mathbf{j} + 6\mathbf{k}$ respectively. The line $l_1$ passes through $A$ and $B$.

May/June 2025

Relative to origin O, the position vectors of points A, B and C are given by $\overrightarrow{OA} = \begin{pmatrix} 1 \\ -4 \\ 2 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} -2 \\ 1 \\ 3 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}$.

May/June 2025

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = i + 2j$, $\overrightarrow{OB} = i + 3j - 2k$ and $\overrightarrow{OC} = 2i - j + 3k$. The line $l$ goes through $B$ and $C$.

May/June 2025

Relative to the origin $O$, the position vectors for $A$, $B$ and $C$ are given by $\overrightarrow{OA} = 2\mathbf{i} - \mathbf{j} - 6\mathbf{k}$, $\overrightarrow{OB} = b\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OC} = -4\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}$.

May/June 2025

Relative to the origin $O$, the points $A$ and $B$ have position vectors $overrightarrow{OA} = i + 2j + 2k$ and $overrightarrow{OB} = 3i + 4j.$ Point $P$ is on the line $AB$, and $OP$ is at right angles to $AB.$

Oct/Nov 2010

Relative to the origin $O$, the position vectors of $A$ and $B$ are $ \overrightarrow{OA} = i + 2j + 2k$ and $\overrightarrow{OB} = 3i + 4j$. Point $P$ is located on the line $AB$, and $OP$ is perpendicular to $AB$.

Oct/Nov 2010

The line $l$ passes through the points $(-5, 3, 6)$ and $(5, 8, 1)$. The plane $p$ is given by $2x - y + 4z = 9$.

Oct/Nov 2010

Relative to the origin $O$, the position vectors of points $A$ and $B$ are $\overrightarrow{OA} = i + 2j + 2k$ and $\overrightarrow{OB} = 3i + 4j$. Point $P$ is on the straight line through $A$ and $B$, with $\overrightarrow{AP} = \lambda \overrightarrow{AB}$.

Oct/Nov 2011

Relative to the origin $O$, the position vectors of points $A$ and $B$ are $overrightarrow{OA} = i + 2j + 2k$ and $overrightarrow{OB} = 3i + 4j$. Point $P$ is on the straight line through $A$ and $B$, and $overrightarrow{AP} = \lambda \overrightarrow{AB}$.

Oct/Nov 2011

The line $l$ is given by $\mathbf{r} = \begin{pmatrix} a \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix}$, with $a$ as a constant. The plane $p$ is defined by $2x - 2y + z = 10$.

Oct/Nov 2011

With origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\overrightarrow{OA} = \begin{pmatrix}3\\-2\\4\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}2\\-1\\7\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1\\-5\\-3\end{pmatrix}$. The plane $m$ is parallel to $OC$ and passes through $A$ and $B$.

Oct/Nov 2012

Taking the origin $O$ as the reference point, the position vectors of $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix}3 \\ -2 \\ 4\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}2 \\ -1 \\ 7\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1 \\ -5 \\ -3\end{pmatrix}$. The plane $m$ is parallel to $OC$ and passes through $A$ and $B$.

Oct/Nov 2012

The equations of the two lines are $\mathbf{r}=\begin{pmatrix}5\\1\\-4\end{pmatrix}+s\begin{pmatrix}1\\-1\\3\end{pmatrix}$ and $\mathbf{r}=\begin{pmatrix}p\\4\\-2\end{pmatrix}+t\begin{pmatrix}2\\5\\-4\end{pmatrix}$, with $p$ being a constant. It is stated that the lines intersect.

Oct/Nov 2012

The diagram depicts three points $A$, $B$ and $C$, with position vectors relative to the origin $O$ given by $\vec{OA} = \begin{pmatrix}2 \\ -1 \\ 2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}0 \\ 3 \\ 1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}3 \\ 0 \\ 4\end{pmatrix}$. Point $D$ is situated on $BC$, between $B$ and $C$, and satisfies $CD = 2DB$.

Oct/Nov 2013

The diagram depicts three points $A$, $B$ and $C$ whose position vectors relative to the origin $O$ are $\vec{OA} = \begin{pmatrix}2\\-1\\2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}0\\3\\1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}3\\0\\4\end{pmatrix}$. The point $D$ lies on $BC$, with $B$ and $C$ as the end points, and satisfies $CD = 2DB$.

Oct/Nov 2013

The two planes are given by $3x - y + 2z = 9$ and $x + y - 4z = -1$.

Oct/Nov 2013

The equation of line $l$ is $\mathbf{r} = 4\mathbf{i} - 9\mathbf{j} + 9\mathbf{k} + \lambda(-2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$. Point $A$ is given by the position vector $3\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}$.

Oct/Nov 2014

The line $l$ is given by equation $\mathbf{r} = 4\mathbf{i} - 9\mathbf{j} + 9\mathbf{k} + \lambda(-2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$. Point $A$ has position vector $3\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}$.

Oct/Nov 2014

The position vectors of points $A$, $B$ and $C$, measured from the origin $O$, are given by $\vec{OA} = \begin{pmatrix}1\\2\\0\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}3\\0\\1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}1\\1\\4\end{pmatrix}$. Plane $m$ is perpendicular to $AB$ and passes through point $C$.

Oct/Nov 2015

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix}1\\2\\0\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}3\\0\\1\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}1\\1\\4\end{pmatrix}$. The plane $m$ is perpendicular to $AB$ and passes through the point $C$.

Oct/Nov 2015

A plane is described by the equation $4x - y + 5z = 39$. A straight line is parallel to the vector $\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$ and goes through the point $A(0, 2, -8)$. This line intersects the plane at point $B$.

Oct/Nov 2015

The two planes are given by $3x + y - z = 2$ and $x - y + 2z = 3$.

Oct/Nov 2016

The equations for two planes are $3x + y - z = 2$ and $x - y + 2z = 3$.

Oct/Nov 2016

The line $l$ is represented by the vector equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + \mathbf{k})$.

Oct/Nov 2016

The equations for the two lines $l$ and $m$ are $\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$, in that order.

Oct/Nov 2017

The planes $p$ and $q$ are defined by the equations $x + y + 3z = 8$ and $2x - 2y + z = 3$, respectively.

Oct/Nov 2017

The pair of lines $l$ and $m$ is given by $\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})$ and $\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$ respectively.

Oct/Nov 2017

The planes $m$ and $n$ are given by the equations $3x + y - 2z = 10$ and $x - 2y + 2z = 5$, respectively. The line $l$ is given by $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

Oct/Nov 2018

The line $l$ is represented by $\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})$. The plane $p$ is represented by $(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0$. The line $l$ meets the plane $p$ at the point $A$.

Oct/Nov 2018

The planes $m$ and $n$ are given by the equations $3x + y - 2z = 10$ and $x - 2y + 2z = 5$ respectively. The line $l$ is represented by $\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})$.

Oct/Nov 2018

The two lines $l$ and $m$ are given by the equations $\mathbf{r} = a\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 3\mathbf{k})$ and $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k})$ respectively, with $a$ as a constant. The lines are known to intersect.

Oct/Nov 2019

The line $l$ is given by $\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 3\mathbf{k})$. The plane $p$ is described by $2x + y - 3z = 5$.

Oct/Nov 2019

Plane $m$ is given by $x + 4y - 8z = 2$. Plane $n$ is parallel to $m$ and goes through $P$, whose coordinates are $(5, 2, -2)$.

Oct/Nov 2019

The two lines are given by $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k})$, where $a$ is a constant.

Oct/Nov 2020

Taking origin $O$ as the reference point, the position vectors of the points $A$, $B$, $C$ and $D$ are $\vec{OA} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix}$, $\vec{OC} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ and $\vec{OD} = \begin{pmatrix} 3 \\ 2 \\ 3 \end{pmatrix}$.

Oct/Nov 2020

The two lines are given by the equations $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 2\mathbf{i} + \mathbf{j} - \kappa + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k})$, where $a$ is a constant.

Oct/Nov 2020

The two lines $l$ and $m$ are represented by $\mathbf{r} = 3\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + s(4\mathbf{i} - \mathbf{j} + 3\mathbf{k})$ and $\mathbf{r} = \mathbf{i} - \mathbf{j} - 2\mathbf{k} + t(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$ respectively.

Oct/Nov 2021

With regard to the origin $O$, the position vectors for the points $A$ and $B$ are $vec{OA} = \begin{pmatrix}1\\2\\-1\end{pmatrix}$ and $\vec{OB} = \begin{pmatrix}0\\3\\1\end{pmatrix}$.

Oct/Nov 2021

The diagram shows $OABCD$ as a pyramid with vertex $D$. Its horizontal base $OABC$ is a square with side $4$ units. Edge $OD$ is vertical, and $OD = 4$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$ respectively. $M$ is the midpoint of $AB$, and $N$ is a point on $CD$ such that $DN = 3NC$.

Oct/Nov 2021

From the diagram, $OABCD$ is a solid shape with $OA = OB = 4$ units and $OD = 3$ units. The edge $OD$ is vertical, $DC$ runs parallel to $OB$, and $DC = 1$ unit. The base $OAB$ is horizontal, while $ngle AOB = 90^07$. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OD$ respectively. $M$ is the midpoint of $AB$, and $N$ lies on $BC$ so that $CN = 2NB$.

Oct/Nov 2022

Measured from the origin $O$, the position vectors of $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}$ and $\vec{OC} = \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}$.

Oct/Nov 2022

Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix}0 \\ 5 \\ 2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}4 \\ -3 \\ -2\end{pmatrix}$. The midpoint of $AC$ is $M$ and $N$ is a point on $BC$, between $B$ and $C$, such that $BN = 2NC$.

Oct/Nov 2022