The following diagram shows the paralleibvcl9mezrd 1u7s ( 2/* qwz-llogram ABCD.

Let E denote the midpoint of [B D], $\mathbf{p}=\overrightarrow{\mathrm{BA}} $ and $ \mathbf{q}=\overrightarrow{\mathrm{BC}} $. Find each of the following vectors in terms of $\mathbf{p}$ and $\mathbf{q}$ .
1. $\overrightarrow{\mathrm{AC}}$ ;  (代数式) 
2. $\overrightarrow{\mathrm{BE}}$ ;  (代数式) 
3. $\overrightarrow{\mathrm{AE}}$ .  (代数式) 



[/B D]

  Let $\mathbf{u}=6 \mathbf{i}-2 \mathbf{j}-\mathbf{k}$ and $\mathbf{v}=3 \mathbf{i}+5 \mathbf{j}+3 \mathbf{k}$ .
1. Find:
1. $\mathbf{u}+\mathbf{v}$ ;  (代数式) 
2. $\mid \mathbf{u}$ ;;   
3. $|\mathbf{v}|$ .   
2. Find $\mathbf{u} \cdot \mathbf{v} $.   
3. Find the angle between $\mathbf{u} $ and $\mathbf{v}$ . $\theta \approx$    $^{\circ}$

  $\text { In the following diagram, } \overrightarrow{\mathrm{CA}}=\mathbf{p}, \overrightarrow{\mathrm{CB}}=\mathbf{q} \text { and } \overrightarrow{\mathrm{AD}}=\frac{1}{2} \overrightarrow{\mathrm{AB}} \text {. }$



Express each of the following vectors in terms of $ \mathbf{p}$ and $\mathbf{q} $.
1. $\overrightarrow{\mathrm{AB}}$  (代数式) 
2. $\overrightarrow{\mathrm{CD}}$ .  (代数式) 

Let $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}4 \\ 2 \\ -1\end{array}\right)$ and $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}3 \\ -2 \\ 8\end{array}\right) $.
1. Find $\overrightarrow{\mathrm{BC}} $.
2. Find the unit vector in the direction of $\overrightarrow{\mathrm{AB}} $.
3. Show that $\overrightarrow{\mathrm{AB}}$ is perpendicular to $\overrightarrow{\mathrm{AC}}$ .

  Let $\mathbf{u}=5 \mathbf{i}+2 \mathbf{j}-\mathbf{k} $ and $\mathbf{v}=2 \mathbf{i}-3 \mathbf{j}+n \mathbf{k} $ where $ n \in \mathbb{R}$ .
1. Given that $\mathbf{u}$ is perpendicular to $\mathbf{v} $, find the value of n .   

Let $\mathbf{w}=7 \mathbf{i}+p \mathbf{j}-3 \mathbf{k}$ where $p \in \mathbb{R}$ .
2. Given that $|\mathbf{w}|=\sqrt{74}$ , find the possible values for p . ±   

  The points A and B h6 bk3wrq.z m5vave position vectors $\overrightarrow{\mathrm{OA}}=\left(\begin{array}{c}2 \\ 4 \\ 12\end{array}\right)$ and $ \overrightarrow{\mathrm{OB}}=\left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right)$ .
1. Find $\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}$.$\left(\begin{array}{r}
a \\
b \\
c
\end{array}\right)$ a =    b =    c =   
2. Hence find the area of the triangle OAB .   

  The points P and R ho4 xmd 8p4qopd010zwnnbnegnw8wyc- u : :0* uave position vectors $ \overrightarrow{\mathrm{OP}}=\left(\begin{array}{c}2 \\ 0 \\ -1\end{array}\right)$ and $\overrightarrow{\mathrm{OR}}=\left(\begin{array}{c}-2 \\ 2 \\ 1\end{array}\right)$ .
1. Find $\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OR}}$$\left(\begin{array}{l}
a \\
b \\
c
\end{array}\right)$ a =    b =    c =   
2. Hence find the area of the triangle OPR.   

ABCD is a parallelogram wherygt4, - sh vzkw -)*valw*tu8ve $\overrightarrow{\mathrm{AB}}=-2 \mathbf{i}-\mathbf{j}+6 \mathbf{k}$ and $\overrightarrow{\mathrm{AD}}=-4 \mathbf{i}+\mathbf{j}+3 \mathbf{k}$ .
1. Find the area of the parallelogram A B C D .
2. By using a suitable scalar product of two vectors, determine whether $A \hat{A B C}$ is acute or obtuse.

The points A and B are given bymj biv p96i/r- $\mathrm{A}(-1,2,0)$ and $\mathrm{B}(2,4,1)$ .
1. Find a vector equation of the line L passing through the points A and B .

The plane $\Pi$ is defined by the equation x+2 y-2 z=18 .
2. Find the coordinates of the point of intersection of the line L with the plane $\Pi$ .

  Consider the vectors th)1nqv nl*6q $\mathbf{u}=3 \mathbf{i}-\mathbf{j}-4 \mathbf{k} $ and $\mathbf{v}=\mathbf{i}+2 \mathbf{j}$ .
1. Find $\mathbf{u} \times \mathbf{v}$ .  (代数式) 
2. Hence find the Cartesian equation of the plane containing the vectors $\mathbf{u}$ and $ \mathbf{v}$ , and passing through the point P(1,-2,2) . 8x-y4+z7=   

  Consider the vectorsz k(8x/oe eezgk01ia4 $\mathbf{a}=2 \mathbf{i}+\mathbf{j}-4 \mathbf{k}$ and $ \mathbf{b}=\mathbf{i}-2 \mathbf{j}+5 \mathbf{k}$ .
1. Find $ \mathbf{a} \times \mathbf{b}$ .  (代数式) 
2. Hence find the Cartesian equation of the plane containing the vectors $\mathbf{a}$ and $\mathbf{b}$ and passing through the point P(2,-1,0) . -3x-14y-5x=   

A line $L_{1}$ passes through the points $\mathrm{A}(-2,0,1)$ and $\mathrm{B}(1,4,1) $.
1. Show that $ \overrightarrow{\mathrm{AB}}=\left(\begin{array}{l}3 \\ 4 \\ 0\end{array}\right) $
2. Hence write down:
1. a direction vector for $L_{1}$ ;
2. a vector equation for $L_{1}$ in the form $\mathbf{r}=\mathbf{a}+t \mathbf{b} $.

Another line $L_{2}$ is perpendicular to $ L_{1}$ and has vector equation

$\mathbf{r}=\left(\begin{array}{c}
4 \\
2 \\
-3
\end{array}\right)+s\left(\begin{array}{c}
k \\
-3 \\
1
\end{array}\right), s \in \mathbb{R}$

3. Find the value of k .
4. Show that the point $\mathrm{C}(-4,8,-5)$ lies on $ L_{2}$ .
5. Let D be the point such that A B C D is a parallelogram. Find $ \overrightarrow{\mathrm{OD}}$ .

  Let $\mathbf{u}=3 \mathbf{i}-2 \mathbf{j}+\mathbf{k}$ and $ \mathbf{v}=4 \mathbf{i}-5 \mathbf{j}+6 \mathbf{k}$. The vector $\mathbf{v}-p \mathbf{u} is perpendicular to \mathbf{u}$ . Find the value of p .   

  Consider the vectors given bbcg k:(f r.v6e9bry: ky $\mathbf{u}=\mathbf{i}+3 \mathbf{j}-4 \mathbf{k}$ and $ \mathbf{v}=a \mathbf{i}+c \mathbf{k}$ , where a and c are constants.
It is given that $\mathbf{u} \times \mathbf{v}=3 \mathbf{i}+b \mathbf{j}-6 \mathbf{k}$ , where b is a constant.
1. Find the value of each of the constants a, b and c . a =    b =    c =   
2. Hence find the Cartesian equation of the plane containing the vectors $\mathbf{u}$ and $\mathbf{v}$ and passing through the point $ \mathrm{P}(3,1,0)$ . x-2y-2z =   

A line $L_{1}$ passes through the points $\mathrm{A}(2,1,-1)$ and $\mathrm{B}(5,-5,-4)$ .
1. 1. Show that $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}3 \\ -6 \\ -3\end{array}\right)$ .
2. Write down the vector equation of $ L_{1}$ in the form $\mathbf{r}=\mathbf{p}+t \mathbf{d}$ .

A line $ L_{2}$ has equation $\mathbf{r}=\left(\begin{array}{l}5 \\ 5 \\ 1\end{array}\right)+s\left(\begin{array}{c}-3 \\ 4 \\ 2\end{array}\right), s \in \mathbb{R} $.
2. Find the acute angle between $L_{1}$ and $ L_{2}$ .
3. The lines $L_{1}$ and $ L_{2}$ intersect at point C . Find $\overrightarrow{\mathrm{OC}}$ .

  Five equilateral triangles, each-mxoenh g7z/d; uf -okh;a 6r6 with side length 4 cm, are arranged to form a truss bridge model.This is shown in the har6ne k xz6o/;;ou--gm hf7dfollowing diagram.



The vectors $ \mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are shown on the diagram.
Find $\mathbf{a} \cdot(\mathbf{a}+\mathbf{b}+2 \mathbf{c}) $.   

  The following diagramyz(-kjc/osk hqh5 59l 8whzdk 8e9qd- shows the triangle ABC.

$\text { Given that } \overrightarrow{\mathrm{BA}} \cdot \overrightarrow{\mathrm{BC}}=-6 \text { and }|\overrightarrow{\mathrm{BA}}||\overrightarrow{\mathrm{BC}}|=12 \text {, find the area of triangle } \mathrm{ABC} \text {. }$   

The diagram shows a parallelogr9t4zg f*en dp0am ABCD.


The coordinates of $\mathrm{A}, \mathrm{B} $ and D are $\mathrm{A}(2,3,4), \mathrm{B}(8,5,7)$ and $ \mathrm{D}(4,7,8)$ .
1. 1. Show that $ \overrightarrow{\mathrm{AB}}=\left(\begin{array}{l}6 \\ 2 \\ 3\end{array}\right)$ .
2. Find $\overrightarrow{\mathrm{AD}} .
3. Hence show that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{l}8 \\ 6 \\ 7\end{array}\right)$ .
2. Find the coordinates of point C .
3. 1. Find $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AD}}$ .
2. Hence find B $\hat{A} $D .
4. Hence, or otherwise, find the area of the parallelogram.

$\text { Given any two non-zero vectors, } u \text { and } v \text {, show that }|u \times v|^{2}=|u|^{2}|v|^{2}-(u \cdot v)^{2} \text {. }$

$\text { Given that } 2(\mathbf{a} \times \mathbf{b})=-3(\mathbf{b} \times \mathbf{c}) \text {, prove that } 2 \mathbf{a}-3 \mathbf{c}=t \mathbf{b} \text { where } t \text { is a scalar. }$

  Consider the following t5rp ,y;k nux.jwo planes:

$\begin{array}{ll}
\Pi_{1}: & x-5 y+3 z=10, \\
\Pi_{2}: & 2 x+6 y-z=12 .
\end{array}$


Find the acute angle between $ \Pi_{1}$ and $\Pi_{2}$ , giving your answer correct to the nearest degree.    $^{\circ}$

$\mathrm{O}$,$ \mathrm{A}$, $\mathrm{B}$ and C are distinct points such that $\overrightarrow{\mathrm{OA}}=\mathbf{a}, \overrightarrow{\mathrm{OB}}=\mathbf{b}$ and $\overrightarrow{\mathrm{OC}}=\mathbf{c}$ . It is given that $ \mathbf{a}$ is perpendicular to $ \mathbf{b}$4 and $\mathbf{c}$ .
Prove that $ \mathbf{a}$ is perpendicular to $\overrightarrow{\mathrm{BC}}$ .

The line $ L_{1} $ passes through the points $\mathrm{A}(4,7,-4) $ and $\mathrm{B}(4,6,-3)$ .
1. 1. Show that $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}0 \\ -1 \\ 1\end{array}\right)$ .
2. Write down the vector equation of $ L_{1} $ in the form $\mathbf{r}=\mathbf{p}+t \mathbf{d} $.

Another line $L_{2}$ has equation $\mathbf{r}=\left(\begin{array}{l}2 \\ 1 \\ 4\end{array}\right)+s\left(\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right)$, s $\in \mathbb{R}$ . The lines $L_{1}$ and $L_{2}$ intersect at point C .
2. Find $ \overrightarrow{\mathrm{OC}}$ .
3. Find the acute angle between $ \overrightarrow{\mathrm{OC}} $ and $L_{1} $.

Let A and B be points such thasbm2b.o- ( udat $\overrightarrow{\mathrm{OA}}=\left(\begin{array}{c}3 \\ 1 \\ -2\end{array}\right) $ and $ \overrightarrow{\mathrm{OB}}=\left(\begin{array}{c}1 \\ -2 \\ -1\end{array}\right) $.
1. Show that

Let C and D be points such that A B C D is a rectangle.
2. Given that $\overrightarrow{\mathrm{AD}}=\left(\begin{array}{c}q \\ -4 \\ -6\end{array}\right)$ , show that q=3 .
3. Find the coordinates of point C .
4. Find the area of the rectangle A B C D .

  Let $ \mathbf{u}=\left(\begin{array}{c}1 \\ 3 \\ -2\end{array}\right)$ and $ \mathbf{v}=\left(\begin{array}{c}k \\ 2 \\ -4\end{array}\right)$ where k>0 . The angle between$ \mathbf{u}$ and $\mathbf{v} $ is $ \frac{\pi}{4}$ . Find the value of k .≈   

  Two points A and B have coordinates (2,3,1) and (5,9,3) resp r/ yyh;y+cd2 bd8t1zaectively.
1. 1. Find $\overrightarrow{\mathrm{AB}}$.  (代数式) 
2. Find $|\overrightarrow{\mathrm{AB}}|$ .   

Let $\overrightarrow{\mathrm{AC}}=4 \mathbf{i}+2 \mathbf{j}-4 \mathbf{k}$ .
2. Find the angle between $\overrightarrow{\mathrm{AB}} $ and $ \overrightarrow{\mathrm{AC}}$. ≈   
3. Find the area of triangle A B C .≈   
4. Hence, or otherwise, find the shortest distance from C to the line passing through A and B .≈   

In this question, dista)m f3ih.gm29c8v/ cjwd o/frn nce is in metres.
Jack and John are flying airplanes in a straight line at a constant speed.
Jack's airplane passes through a point P . Its position, t seconds after it passes through P , is given by $\mathbf{r}_{1}=\left(\begin{array}{l}5 \\ 8 \\ 2\end{array}\right)+t\left(\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right), t \in \mathbb{R}$ .
1. 1. Write down the coordinates of P .
2. Find the speed of Jack's airplane in $\mathrm{ms}^{-1} $.
2. After six seconds, Jack's airplane passes through a point Q .
1. Find the coordinates of Q .
2. Find the distance the airplane has travelled during the six seconds.

John's airplane passes through a point R . Its position, s seconds after it passes through R , is given by $ \mathbf{r}_{2}=\left(\begin{array}{l}4 \\ 4 \\ 5\end{array}\right)+s\left(\begin{array}{c}-1 \\ 5 \\ 3\end{array}\right), s \in \mathbb{R}$ .
3. Find the coordinates where the two airplanes intersect.
4. Is Jack or John flying faster? Justify your answer.

  Let $\mathbf{u}=\left(\begin{array}{c}2 \\ -1 \\ 5\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}m \\ n \\ 0\end{array}\right)$ . Given that $\mathbf{v}$ is a unit vector perpendicular to $ \mathbf{u}$ , find the possible values of m and the possible values of n .m =    n =    m =   n =   

  In this question, distances are measured in kilomnf+erwp0* u:( ed /tqwetres.
Two boats, A and B , are observed from an origin O . Relative to O , their position vectors at time t hours after midday are given by

$\begin{array}{l}
\mathbf{r}_{\mathrm{A}}=\binom{-4}{3}+t\binom{4}{3} \\
\mathbf{r}_{\mathrm{B}}=\binom{-2}{9}+t\binom{5}{0}
\end{array}
$

Find the minimum distance between the two boats.   

  Consider the lines Lbqr-xy7(+kd zju qm ;9_{1} and L_{2} defined by

$L_{1}: \mathbf{r}=\left(\begin{array}{c}
-6 \\
10 \\
c
\end{array}\right)+\lambda\left(\begin{array}{c}
3 \\
-3 \\
1
\end{array}\right) \text { and } L_{2}:-\frac{x}{2}=y-1=\frac{z-13}{3}$

where $\lambda \in \mathbb{R}$ and c is a constant. Given that the lines intersect at a point A ,
1. find the value of c ;   
2. determine the coordinates of A . (a,b,c) a =    b =    c =   

The position vectors of the points A, B and C are a, b and c, respectively,la2q( s7b0t 68uqpfokr8om 8+ fv,jroxw fy59 relative to the origin 8uo mr9s(8 tao2r q q lw+8ofb5y0pfxfv,j6k 7O.

$\text { Given that } \mathbf{a} \text { is perpendicular to } \overrightarrow{\mathrm{BC}} \text { and } \mathbf{b} \text { is perpendicular to } \overrightarrow{\mathrm{CA}} \text {, prove that } \mathbf{c} \text { is perpendicular to } \overrightarrow{\mathrm{AB}} \text {. }$

Two lines L_{1} and L_{2} are re2veu nrv+k3 pd )vj.+hpresented by the vector equations:

$\begin{aligned}
L_{1}: & \mathbf{r}_{1}=\left(\begin{array}{l}
2 \\
1 \\
4
\end{array}\right)+t\left(\begin{array}{l}
0 \\
1 \\
2
\end{array}\right), t \in \mathbb{R} \\
L_{2}: & \mathbf{r}_{2}=\left(\begin{array}{c}
-1 \\
-3 \\
11
\end{array}\right)+s\left(\begin{array}{l}
1 \\
2 \\
k
\end{array}\right), s \in \mathbb{R}
\end{aligned}$

The lines $L_{1} $ and $ L_{2}$ are perpendicular to each other.
1. Show that k=-1 .

The lines L_{1} and L_{2} intersect at the point A .
2. Find $\overrightarrow{\mathrm{OA}}$ .

Let $ \overrightarrow{\mathrm{OB}}=\left(\begin{array}{l}2 \\ 2 \\ 7\end{array}\right)$ and $\overrightarrow{\mathrm{BC}}=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$
3. 1. Find $\overrightarrow{\mathrm{BA}}$.
2. Hence find the angle CBA.

A line L passes through the poin nndu ,;(w9evqn7 dg8uts$\mathrm{A}(6,5,0)$ and $ \mathrm{B}(8,6,1)$ .
1. 1. Show that $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{l}2 \\ 1 \\ 1\end{array}\right)$ .
2. Write down a vector equation of L in the form $\mathbf{r}=\mathbf{p}+t \mathbf{d}$ .

The line L also passes through the point $\mathrm{C}(2, b,-2)$ .
2. Find the value of b .

The point D has coordinates $\left(p^{2}, 2,5 p\right)$ .
3. Given that $ \overrightarrow{\mathrm{CD}}$ is perpendicular to L , find the possible values of p .

Consider the points kj,f(vwlx p/:* d l*ry$\mathrm{A}(3,2,-5) $ and $ \mathrm{B}(-3,6,-5)$ .
1. Find $\overrightarrow{\mathrm{AB}}$ .

Let C be a point such that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{l}3 \\ 0 \\ 2\end{array}\right)$ .
2. Find the coordinates of C .

The line L passes through B and is parallel to $\overrightarrow{\mathrm{AC}}$ .
3. Write down a vector equation of L in the form $\mathbf{r}=\mathbf{p}+t \mathbf{d}$ .
4. Given that $|\overrightarrow{\mathrm{AB}}|=k|\overrightarrow{\mathrm{AC}}|$ , find k .

The point D lies on L such that $|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BD}}| $.
5. Find the possible coordinates of D .

  The line L has equ14qn qm +/j;+cqrgdtnation $\mathbf{r}=\left(\begin{array}{c}4 \\ -1 \\ 0\end{array}\right)+t\left(\begin{array}{c}1 \\ 1 \\ -1\end{array}\right), t \in \mathbb{R}$ .
The point A has coordinates (4,8,-3) . The point B lies on L such that $\overrightarrow{\mathrm{AB}}$ is perpendicular to L .
1. Find $\overrightarrow{\mathrm{OB}}$ .$\left(\begin{array}{l}
a \\
b \\
c
\end{array}\right)$ a =    b =    c =   
2. Find the shortest distance from A to L .   
3. Find the coordinates of the reflection of the point A in the line L . (a,b,c) a =    b =    c =   

  In this question, distances are melfc+ 3 s(0b, /q7ls o8oimwikeasured in kilometres.
Two tankers, A and B , are observed from an origin O . Relative to O , their position vectors at time t hours after 6:00pm are given by

$\begin{array}{l}
\mathbf{r}_{\mathrm{A}}=\binom{1}{1}+t\binom{7}{4} \\
\mathbf{r}_{\mathrm{B}}=\binom{6}{9}+t\binom{5}{3}
\end{array}
$

Find the time when the distance between the two tankers is at a minimum. a : b pm a =    b =   

Consider two vectors b( :hvbnb3vybne9*w(lpxs 7 .ale.k1 $\mathbf{u}$ and \mathbf{v} such that $\mathbf{u}=\binom{-6}{8} $ and $|\mathbf{v}|=20$ .
1. Find the possible range of values for $|\mathbf{u}+\mathbf{v}|$.
2. Given that $\mathbf{v}=k \mathbf{u} $ for some $k \in \mathbb{R}$ , find $\mathbf{v}$ when $|\mathbf{u}+\mathbf{v}| $ is a minimum.
3. Find the vector $\mathbf{w}=\binom{a}{b}$ such that a, b $\in \mathbb{R}^{+},|\mathbf{w}|=|\mathbf{v}|$ and $\mathbf{w}$ is perpendicular to $ \mathbf{u}$ .

  $\text { Find the Cartesian equation of the plane } \Pi \text { containing the points } \mathrm{A}(3,-1,3) \text { and } \mathrm{B}(4,1,-1) \text {, and perpendicular to the plane } 2 x-5 y+z=10 \text {. }$2 x+y+z=   

  The lines L_{1} and L_{2} are defined as

$\begin{array}{l}
L_{1}: \frac{x+2}{-3}=\frac{y-1}{4}=\frac{z+6}{2} \\
L_{2}: \quad \frac{x+2}{6}=\frac{y-1}{2}=\frac{z+6}{-4}
\end{array}$


The plane $ \Pi$ contains both $ L_{1} $ and $ L_{2} $.
1. Find the Cartesian equation of $\Pi$ .2 x+3 z=   

The line$ L_{3} $ is passing through the point $ \mathrm{P}(1,-2,5)$ and perpendicular to$ \Pi $.
2. Find the coordinates of the point where$ L_{3}$ intersects $\Pi$.(a,b,c) a =    b =    c =   

The points A and B hawr4xh ea49-b eve coordinates $ \mathrm{A}(1,0,4)$ and $\mathrm{B}(2,2,3)$ relative to an origin O .
1. 1. Find $\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OB}}$ .
2. Determine the area of the triangle O A B , giving your answer correct to two decimal places.
3. Find the Cartesian equation of the plane OAB .
2. 1. Find the vector equation of the line L_{1} containing the points A and B .
The line $ L_{2} $ has vector equation $\mathbf{r}=\left(\begin{array}{c}0 \\ 12 \\ -9\end{array}\right)+s\left(\begin{array}{c}-3 \\ 1 \\ -4\end{array}\right)$, s $\in \mathbb{R}$ .
2. Determine whether the lines $L_{1}$ and $L_{2} $ are parallel, skew or intersecting. If they intersect, find the coordinates of the point of intersection.

  The plane $\Pi$ has the Cartesian equation x-y+p z=10 , where $ p \in \mathbb{R}$, p<0 .
The line L has the vector equation $\mathbf{r}=\left(\begin{array}{l}2 \\ 5 \\ 4\end{array}\right)+t\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$,$ t \in \mathbb{R}$ .
The acute angle between $\Pi$ and L is $30^{\circ}$ . Find the value of p .   

The position vectors of the points A, B and C are a,b and c, respectijia 3 1je:z38amt9ao zvely, relative to an origin O.The following diagram shows the t3a9z 3a:8jjetm zoai1 riangle ABC and points N, R, S and T.


N is a point on $ [\mathrm{AB}] $ such that $ \overrightarrow{\mathrm{AN}}=\frac{3}{7} \overrightarrow{\mathrm{AB}}$ .
R is a point on $[\mathrm{BC}]$ such that $\overrightarrow{\mathrm{BR}}=\frac{2}{5} \overrightarrow{\mathrm{BC}} $.
S is a point on $[\mathrm{CA}] $ such that $\overrightarrow{\mathrm{CS}}=\frac{2}{5} \overrightarrow{\mathrm{CA}}$ .
T is a point on $[\mathrm{RS}] $ such that $ \overrightarrow{\mathrm{RT}}=\frac{2}{3} \overrightarrow{\mathrm{RS}}$ .
1. 1. Express $\overrightarrow{\mathrm{AN}}$ in terms of $\mathbf{a} $ and $\mathbf{b}$ .
2. Hence show that $ \overrightarrow{\mathrm{CN}}=\frac{4}{7} \mathbf{a}+\frac{3}{7} \mathbf{b}-\mathbf{c}$ .
2. 1. Express $ \overrightarrow{\mathrm{RC}}$ in terms of $\mathbf{b}$ and $\mathbf{c} $ and $ \overrightarrow{\mathrm{CS}} $ in terms of $ \mathbf{a}$ and $\mathbf{c}$ .
2. Hence show that $\overrightarrow{\mathrm{RT}}=\frac{4}{15} \mathbf{a}-\frac{6}{15} \mathbf{b}+\frac{2}{15} \mathbf{c} $.
3. Prove that T lies on $[\mathrm{CN}]$ .

  The plane $\Pi$ has the Cartesian equation x+y+3 z=5 . The line L has the vector equation $\mathbf{r}=\left(\begin{array}{c}2 \\ 3 \\ -1\end{array}\right)+\lambda\left(\begin{array}{c}-1 \\ 1 \\ p\end{array}\right), \lambda, p \in \mathbb{R}$ . The acute angle between the line L and the plane $\Pi $ is $60^{\circ}$ . Find the possible values of p . ±   

Consider the planes yj bbfa 6tckm c,96/u,$\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$ given by the following equations:

$\begin{array}{lr}
\Pi_{1}: & 2 x+y-z=-3 \\
\Pi_{2}: & x+5 y-5 z=-6 \\
\Pi_{3}: & 3 x+5 y-5 z=-7
\end{array}
$
1. Show that the three planes do not intersect.

It is given that the point $\mathrm{Q}(-1,-1,0)$ lies on both $\Pi_{1}$ and $\Pi_{2} $.
Let $\ell$ be the line of intersection of $\Pi_{1}$ and $\Pi_{2} $.
2. Find a vector expression for $ \ell$ .
3. Show that $\ell$ is parallel to plane $\Pi_{3}$ .
4. Hence or otherwise, find the distance between $\ell$ and $\Pi_{3}$ Express your answer in the form $\frac{p}{\sqrt{q}}$ , where p, q$\in \mathbb{Z}$ .

The points $\mathrm{A}, \mathrm{B}$ and C have the following position vectors with respect to an origin O .

$\begin{array}{l}
\overrightarrow{\mathrm{OA}}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k} \\
\overrightarrow{\mathrm{OB}}=3 \mathbf{i}-2 \mathbf{j}+\mathbf{k} \\
\overrightarrow{\mathrm{OC}}=3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}
\end{array}$

1. Find the vector equation of the line (A B) .
2. Determine whether the lines (AB) and (OC) are parallel, skew or intersecting.
3. Find the Cartesian equation of the plane $\Pi_{1}$ , that passes through the point A and is perpendicular to $ \overrightarrow{\mathrm{OC}}$.
4. Show that the line (A B) lies in the plane $\Pi_{1}$ .

The plane $ \Pi_{2} $ contains the points $\mathrm{O}, \mathrm{A}$ and C and the plane $\Pi_{3} $ contains the points $\mathrm{O}, \mathrm{B}$ and C .
5. Verify that $4 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}$ is perpendicular to the plane $ \Pi_{2} $.
6. Find a vector perpendicular to the plane $\Pi_{3}$ .
7. Find the acute angle between the planes $\Pi_{2} $ and $\Pi_{3}$ , giving your answer correct to the nearest degree.

The plane $\Pi$ has equation 2 x-y+z=15 .
The line L is perpendicular to the plane $\Pi $and has a vector equation

$\mathbf{r}=\left(\begin{array}{c}
1 \\
-p \\
3
\end{array}\right)+\lambda\left(\begin{array}{c}
-4 \\
2 \\
p
\end{array}\right), \quad \lambda \in \mathbb{R}$ .

1. Determine the value of p .
2. Find the coordinates of the point where L meets the plane $\Pi$ .
3. The point P(3,-1,2) does not lie on the plane $\Pi$ . Find:
1. $\overrightarrow{\mathrm{PA}}$ if A is a general point on L ;
2. $\lambda such that \overrightarrow{\mathrm{PA}} $ is perpendicular to L ;
3. the coordinates of the foot of the perpendicular from P to L ;
4. the shortest distance from P to L .
4. Given two planes 2 x-y+z=15 and x+3 y-2 z=8 , find the equation of a third plane which is perpendicular to both of these planes and which cuts the x -axis at -5 .

The plane $\Pi$ has equation 2 x-y+z=15 .
The line L is perpendicular to the plane $\Pi $and has a vector equation

$\mathbf{r}=\left(\begin{array}{c}
1 \\
-p \\
3
\end{array}\right)+\lambda\left(\begin{array}{c}
-4 \\
2 \\
p
\end{array}\right), \quad \lambda \in \mathbb{R}$ .

1. Determine the value of p .
2. Find the coordinates of the point where L meets the plane $\Pi$ .
3. The point P(3,-1,2) does not lie on the plane $\Pi$ . Find:
1. $\overrightarrow{\mathrm{PA}}$ if A is a general point on L ;
2. $\lambda such that \overrightarrow{\mathrm{PA}} $ is perpendicular to L ;
3. the coordinates of the foot of the perpendicular from P to L ;
4. the shortest distance from P to L .
4. Given two planes 2 x-y+z=15 and x+3 y-2 z=8 , find the equation of a third plane which is perpendicular to both of these planes and which cuts the x -axis at -5 .

Two lines have vector equations givx- ggv 5h0(vcpqa uye;3l+ uu(en by

$L_{1}: \mathbf{r}_{1}=\left(\begin{array}{c}
3 \\
-1 \\
2
\end{array}\right)+t\left(\begin{array}{l}
1 \\
2 \\
1
\end{array}\right), t \in \mathbb{R} . L_{2}: \mathbf{r}_{2}=\left(\begin{array}{c}
4 \\
-1 \\
0
\end{array}\right)+s\left(\begin{array}{l}
1 \\
3 \\
1
\end{array}\right), s \in \mathbb{R} $.

Point A is the point on $L_{1}$ that is closest to the origin.
1. Find the coordinates of A .
2. Find the shortest distance between $L_{1}$ and $ L_{2}$ .

Two planes have equations n6ops i6p rfj9,m8dd i,*wo d.m5bpb3

$\Pi_{1}$: 2 x+4 y+z=9 $\text { and } \Pi_{2}$: 2 x+y-z=1 .

1. Find the cosine of the angle between the two planes, giving your answer in the form $\frac{\sqrt{p}}{q}$ where p, q $\in \mathbb{Z}^{+}$ .

Let L be the line of intersection of the two planes.
2. 1. Show that L has direction $5 \mathbf{i}-4 \mathbf{j}+6 \mathbf{k}$ .
2. Show that the point P(0,2,1) lies on both planes.
3. Write down the vector equation of L .

Q is the point on $\Pi_{1} $ with coordinates (a, 1, b) .
3. Given that the vector $\overrightarrow{\mathrm{PQ}} $ is perpendicular to L , find the value of a and the value of b .
4. Show that $\mathrm{PQ}=\sqrt{33}$ .

The point R lies on L and $\mathrm{PQ} \mathrm{Q}=30^{\circ} $.
5. Find the coordinates of the two possible positions of R.

A line L has equations :tdu9h,2 ix7dr*oked*w;th9d i*ud y$ \frac{x-a}{2}=\frac{y-1}{b}=-\frac{z-1}{2}$ where a, b $\in \mathbb{Z}$ .
A plane $ \Pi $ has equation 2 x+3 y-z=6 .
1. Show that L is not perpendicular to $\Pi$ .
2. Given that L lies in the plane $\Pi $, find the value of a and the value of b .

Consider the different case where the acute angle between L and $ \Pi$ is $\alpha $ where $ \alpha=\arcsin \left(\frac{3}{\sqrt{14}}\right)$.
3. 1. Show that b=1 .
2. Given that L intersects $ \Pi$ at z=-3 , find the value of a .

Two planes have equation *+pi hsgh 1v)hc,*nzs. u p3.lr-cglbs

$\Pi_{1}$: 2 x-2 y+z=1 $\text { and } \Pi_{2}: x-4 y-z=5 $.

1. Find the cosine of the angle between the two planes, giving your answer in the form $\frac{\sqrt{p}}{q}$ where p, q$\in \mathbb{Z}^{+} $.

Let L be the line of intersection of the two planes.
2. 1. Show that L has direction $ 2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$.
2. Show that the point A(4,1,-5) lies on both planes.
3. Write down the vector equation of L .

B is the point on $ \Pi_{1}$ with coordinates (a, b, 3) .
3. Given that the vector $\overrightarrow{\mathrm{AB}}$ is perpendicular to L , find the value of a and the value of b .
4. Show that $ \mathrm{AB}=12$ .

The point C lies on L and $\mathrm{AB} \mathrm{B}=60^{\circ}$ .
5. Find the coordinates of the two possible positions of C .

The equations of the lqyu2t0 vswva. iv)j14r*n(r yines $ L_{1}$ and $ L_{2}$ are

$\begin{array}{l}
L_{1}: \mathbf{r}_{1}=\left(\begin{array}{l}
2 \\
3 \\
1
\end{array}\right)+s\left(\begin{array}{c}
4 \\
-2 \\
4
\end{array}\right), s \in \mathbb{R} \\
L_{2}: \quad \mathbf{r}_{2}=\left(\begin{array}{l}
1 \\
3 \\
1
\end{array}\right)+t\left(\begin{array}{c}
-2 \\
1 \\
2
\end{array}\right), t \in \mathbb{R}
\end{array}$

1. Show that the lines $L_{1}$ and $ L_{2}$ are skew.
2. Find the acute angle between $ L_{1} $ and $L_{2} $, giving your answer correct to the nearest degree.
3. 1. Find a vector perpendicular to both lines.
2. Hence determine an equation of the line $ L_{3}$ that is perpendicular to both $ L_{1} $ and $ L_{2} $ and intersects both lines.