Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

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.

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

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

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

Find the vector equation of the line (BC).

Consider the vectors a = $\left(\begin{array}{c}3\\ 2p\end{array}\right)$ and b = $\left(\begin{array}{c}p+1\\ 8\end{array}\right)$.

Find the possible values of p for which a and b are parallel.

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 the area of triangle PQR.

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

Find and simplify an expression for a • b in terms of $t$.

Find the value of $p$.

Find a vector equation for L₁.

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

Find $\mathit{q}$ such that $\left|\mathit{q}\right|=\left|\mathit{b}\right|$ and $\mathit{q}$ is perpendicular to $\mathit{a}$.

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

Let $D\left(t\right)$ represent the distance between airplane $A$ and airplane $B$ for $0\le t\le 2.5$.

Find the minimum value of $D\left(t\right)$.

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 $\Pi$.

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.

perpendicular.

Find $\overrightarrow{\text{AB}}$.

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

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

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

The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.

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

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

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

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

Find a vector equation for $L$.

Given any two non-zero vectors, $\mathit{a}$ and $\mathit{b}$, show that ${\left|\mathit{a}\times \mathit{b}\right|}^2={\left|\mathit{a}\right|}^2{\left|\mathit{b}\right|}^2-{\left(\mathit{a}·\mathit{b}\right)}^2$.

Find the angle between PQ and PR.

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

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

Find $\overrightarrow{AB}$.

Let a = $\left(\begin{array}{c}2\\ k\\ -1\end{array}\right)$ and b = $\left(\begin{array}{c}-3\\ k+2\\ k\end{array}\right)$, $k\in \mathbb{R}$.

Given that a and b are perpendicular, find the possible values of $k$.

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

Find the value of $y$.

Find $\overrightarrow{\text{PQ}}$.

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

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 $\text{P}$.

Hence or otherwise find the shortest distance from R to the line through P and Q.

Given that c = a + 2b, find c.

Find the value of $k$.

The magnitudes of two vectors, u and v, are 4 and $\sqrt{3}$ respectively. The angle between u and v is $\frac{\pi }{6}$.

Let w = u − v. Find the magnitude of w.

Write down the value of angle OBA.

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

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

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

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 $L$.

Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .

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

parallel.

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

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

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

By using a suitable scalar product of two vectors, determine whether $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is acute or obtuse.

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

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

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 $\mathit{r}=\left(\begin{array}{c}0\\ -2\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)$.