Given that $\left|\mathit{a}+\mathit{b}\right|$ is a minimum, find $\mathit{p}$.

Find the distance between the two submarines at this time.

Find the value of t when submarine B passes through P.

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.

Find, in terms of a and b $\overrightarrow{\text{OF}}$.

Explain why ABCD is a parallelogram.

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

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

Find the value of t when the two submarines are closest together.

$\overrightarrow{\text{OA}}\bullet \overrightarrow{\text{OC}}$.

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

Write down a direction vector for $L_3$.

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

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

Find an expression for the distance between the two submarines in terms of t.

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 $L_2$.

Find the area of triangle ABC.

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

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

$\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{OC}}$.

Find the possible range of values for $\left|\mathit{a}+\mathit{b}\right|$.

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 $L_1$.

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\lambda$;

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

perpendicular.

Find $\overrightarrow{\text{AB}}$.

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

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 $\overrightarrow{\text{AC}}$.

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\mu$.

Hence or otherwise, find the area of the triangle ABC.

Given that $\text{A}\hat{\text{O}}\text{C}=\text{B}\hat{\text{O}}\text{C}$, show that $k=7$.

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

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

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

Find $\overrightarrow{AB}$.

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

Find the value of $y$.

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

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.

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

Let $\overrightarrow{\text{AC}}=\left(\begin{array}{c}3\\ 0\\ 0\end{array}\right)$. Find $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

Find the coordinates of X, Y and Z.

Find the area of the parallelogram ABCD.

Deduce an expression for $\overrightarrow{\text{CD}}$ in terms of a and b only.

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 $L$.

Given that area $\mathrm{\Delta }\text{OAB}=k(\text{area }\mathrm{\Delta }\text{CAD})$, find the value of $k$.

parallel.

Show that $\mu =\frac{1}{13}$, and find the value of $\lambda$.

Calculate the area of triangle $\text{AOC}$.

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

Show that submarine B travels in the same direction as originally planned.

Find, in terms of a and b $\overrightarrow{\text{AF}}$.

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

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 $\text{C}$.

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

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