Verify that the point $\text{P}(1, -2, 0)$ lies on both $\prod _1$ and $\prod _2$.

Hence or otherwise, find the distance between $P_1$ and $P_2$ two seconds after they leave A.

Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.

Hence or otherwise, find one point on $L_2$ which is $\sqrt{5}$ units from C.

Explain why ABCD is a parallelogram.

Find the Cartesian equation of $\mathrm{\Pi }$.

Write down $\overrightarrow{\text{PA}}$ and $\overrightarrow{\text{PB}}$.

Find a vector perpendicular to the plane Π${}_3$.

Find how many sets of four points can be selected which can form the vertices of a quadrilateral.

Find the vector equation of the line (BC).

Write down a direction vector for $L_3$.

Write down the value of $\lambda$ corresponding to the point $\text{A}$.

A straight line, $L_{\theta }$, has vector equation r $=\left(\begin{array}{c}5\\ 0\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}5\\ \text{sin}\theta \\ \text{cos}\theta \end{array}\right)\text{, }\lambda \text{, }\theta \in \mathbb{R}$.

The plane ${\Pi }_p$, has equation $x=p\text{, }p\in \mathbb{R}$.

Show that the angle between $L_{\theta }$ and ${\Pi }_p$ is independent of both $\theta$ and $p$.

Find how many sets of three points can be selected which can form the vertices of a triangle.

Find the coordinates of A.

Find the vector equation of the straight line passing through M and normal to the plane $\mathrm{\Pi }$ containing ABCD.

Find a unit vector in the direction of $L_2$.

Show that the line (BC) lies in the plane Π${}_1$.

Find a vector equation of the line that passes through P and Q.

Find the area of triangle ABC.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Find $\cos \mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$.

Find the value of $p$.

Find a vector equation for L₁.

find the value of $a$;

Find a vector equation of $L$, the line of intersection of $\prod _1$ and $\prod _2$.

Find the distance between $L$ and $\prod _3$.

Show that $\overrightarrow{\text{AC}}=\left(\begin{array}{c}8\\ -10\\ -1\end{array}\right)$.

Show that airplane $A$ travels at a greater speed than airplane $B$.

Find a vector equation for L.

Hence, write down a vector equation for $L_1$.

Hence determine the minimum distance, $d_{\text{min}}$, from $\text{D}$ to $\Pi$.

A second line, $L_2$, is parallel to $L_1$ and passes through (1, 2, 3).

Write down a vector equation for $L_2$.

Find, in terms of $b$, the equation of the line L which passes through M and is perpendicular to the plane П.

The vector equation of line $L$ is given by r $=\left(\begin{array}{c}-1\\ 3\\ 8\end{array}\right)+t\left(\begin{array}{c}4\\ 5\\ -1\end{array}\right)$.

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.

Show that the point $(-1, 0, 3)$ lies on $L_1$.

Find the possible values of $a$ when the acute angle between $L_1$ and $L_2$ is $45°$.

Find $\overrightarrow{\text{AB}}$.

Find the Cartesian equation of the plane Π${}_1$, which passes through C and is perpendicular to $\overrightarrow{\mathrm{O}\mathrm{A}}$.

Using vector algebra, show that $\overrightarrow{\text{AD}}=\overrightarrow{\text{BC}}$.

Particle $P_2$ also moves in a straight line. The position of $P_2$ is given by $\mathit{r}=\left(\begin{array}{c}-1\\ 6\end{array}\right)+t\left(\begin{array}{c}4\\ -3\end{array}\right)$.

The speed of $P_1$ is greater than the speed of $P_2$ when $t>q$.

Find the value of $q$.

Find the value of $m$.

Find $\left|\overrightarrow{\text{AB}}\right|$.

$L_3$ passes through the intersection of $L_1$ and $L_2$.

Write down a vector equation for $L_3$.

Find the angle between $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

determine the coordinates of the point of intersection P.

Determine whether or not the lines (OA) and (BC) intersect.

Find the gradient of $L$.

Show that p = 1, q = 1 and r = 4.

Let $\text{C}$ be the point on $l_1$ with coordinates (1, 0, 1) and $\text{D}$ be the point on $l_2$ with parameter $\mu =-2$.

Find the area of the quadrilateral $\text{CDBA}$.

Find $\overrightarrow{\text{AB}}$;

Find the vectors $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Find a vector equation for $L$.

Use a vector method to show that $\text{B}\hat{\text{A}}\text{C}=60°$.

Find $\overrightarrow{AB}$.

Find a vector equation of $L_1$.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}6\\ 8\\ -5\end{array}\right)$

Find the value of $y$.

Find the coordinates of the point of intersection of the line L with the plane Π.

Find the point of intersection of $L_1$ and $L_2$.

The point D has coordinates $(q^2,0,q)$. Given that $\overrightarrow{\text{DC}}$ is perpendicular to $L$, find the possible values of $q$.

Show that the three planes do not intersect.

Find the vector $\overrightarrow{\text{AB}}$ and the vector $\overrightarrow{\text{AC}}$.

Hence find the equation of ${\Pi }_1$, expressing your answer in the form $ax+by+cz=d$, where $a, b, c, d\in \mathrm{\mathbb{Z}}$.

Find the reflection of the point $\text{B}$ in the plane ${\Pi }_3$.

Hence find the vector equation of the line formed when $L$ is reflected in the plane ${\Pi }_3$.

Show that at the point $\text{P}, \lambda =\frac{3}{4}$.

Hence find the coordinates of $\text{P}$.

Find YZ.

Verify that $\text{P}$ is the point of intersection of the two lines.

Write down the value of angle OBA.

Find the acute angle between the planes Π${}_2$ and Π${}_3$.

Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time t hours after midday are given by

r_A = $\left(\begin{array}{c}4\\ 3\end{array}\right)+t\left(\begin{array}{c}5\\ 8\end{array}\right)$

r_B = $\left(\begin{array}{c}7\\ -3\end{array}\right)+t\left(\begin{array}{c}0\\ 12\end{array}\right)$

where distances are measured in kilometres.

Find the minimum distance between the two ships.

Point C (k , 12 , −k) is on L. Show that k = 14.

Find the coordinates of X, Y and Z.

Find the area of the parallelogram ABCD.

The vector $\left(\begin{array}{c}2\\ p\\ 0\end{array}\right)$ is perpendicular to $\overrightarrow{\text{AB}}$. Find the value of p.

Find an expression for the velocity of $P_1$ at time $t$.

Express $\overrightarrow{\text{AB}}$ in terms of $m$.

Show that the Cartesian equation of the plane $\Pi$ that contains the triangle $\text{ABC}$ is $-x+y+z=-2$.

Find a vector equation of the line $L$.

Find a vector equation of the line L passing through the points A and B.

Show that L does not intersect the $y$-axis for any negative value of $b$.

Verify that 2j + k is perpendicular to the plane Π${}_2$.

The lines $L_1$ and $L_1$ intersect at $C(9,13,z)$. Find $z$.

Find $\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{AB}}$.

Find, in terms of $b$, a Cartesian equation of the plane Π containing this triangle.

Consider a unit vector $\mathit{u}$, such that $\mathit{u}=p\mathit{i}-\frac{2}{3}\mathit{j}+\frac{1}{3}\mathit{k}$, where $p>0$.

Point $\text{C}$ is such that $\overrightarrow{\text{BC}}=9\mathit{u}$.

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

Show that $l_1$ and $l_2$ are never perpendicular to each other.

A second line $L_2$, has equation r = $\left(\begin{array}{c}1\\ 13\\ -14\end{array}\right)+s\left(\begin{array}{c}p\\ 0\\ 1\end{array}\right)$.

Given that $L_1$ and $L_2$ are perpendicular, show that $p=2$.

Find $c$.

Find the volume of right-pyramid $\text{ABCD}$.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Show that $l_1$ and $l_2$ do not intersect.

It is given that the lines $L_1$ and $L_2$ have a unique point of intersection, $\text{A}$, when $a\ne k$.

Find the value of $k$, and find the coordinates of the point $\text{A}$ in terms of $a$.

Find the minimum distance between $l_1$ and $l_2$.

The line $L$ is the intersection of ${\Pi }_1$ and ${\Pi }_2$. Verify that the vector equation of $L$ can be written as $\mathit{r}=\left(\begin{array}{c}0\\ -2\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)$.