| 1 | (IB/sl/2018/November/Paper1/q5)
[Maximum mark: 6]
Consider the vectors $a=\left(\begin{array}{c}3 \\ 2 p\end{array}\right)$ 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.
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| 2 | (IB/sl/2017/May/paper ltz2/q2)
[Maximum mark: 7 ]
The vectors $a=\left(\begin{array}{l}4 \\ 2\end{array}\right)$ and $b=\left(\begin{array}{c}k+3 \\ k\end{array}\right)$ are perpendicular to each other.
| (a) | Find the value of $k$. |
| (b) | Given that $c=a+2 b$, find $c$. [3] |
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| 3 | (IB/s1/2019/May/paper1tz1/q2)
[Maximum mark; 6]
A line, $L_{1}$, has equation $\boldsymbol{r}=\left(\begin{array}{c}-3 \\ 9 \\ 10\end{array}\right)+s\left(\begin{array}{l}6 \\ 0 \\ 2\end{array}\right) .$ Point $\mathrm{P}(15,9, c)$ lies on $L_{1}$.
Asecond line, $L_{2}$, is parallel to $L_{1}$ and passes through $(1,2,3)$
| (b) | Write down a vector equation for $L_{2}$. [2] |
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| 4 | $\left(\mathrm{IB} / \mathrm{sl} / 2019 / \mathrm{May} / \mathrm{paper} 1 \mathrm{tz} 2 / \mathrm{q}{2}\right)$
[Maximum mark: 6]
Consider the vectors $a=\left(\begin{array}{c}0 \\ 3 \\ p\end{array}\right)$ and $b=\left(\begin{array}{c}0 \\ 6 \\ 18\end{array}\right)$
Find the value of $p$ for which $a$ and $b$ are
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| 5 | $(\mathrm{IB} / \mathrm{sl} / 2019 / \mathrm{May} /$ paper $2 \mathrm{tz} 2 / \mathrm{q} 7)$
[Maximum mark. 6]
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 $\mathrm{P}$ is the point on $L$ that is closest to the origin. Find the coordinates of $\mathrm{P}$.
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| 6 | (IB/s1/2018/November/Paper2/q8)
[Maximum mark: 16]
Consider the points $\mathrm{A}(-3,4,2)$ and $\mathrm{B}(8,-1,5)$.
| (a) | (i) Find $\overrightarrow{\mathrm{AB}}$.
(ii) Find $|\overrightarrow{\mathrm{AB}}|$. |
A line $L$ has vector equation $r=\left(\begin{array}{c}2 \\ 0 \\ -5\end{array}\right)+t\left(\begin{array}{c}1 \\ -2 \\ 2\end{array}\right)$. The point $\mathrm{C}(5, y, 1)$ lies on line $L$.
| (b) | (i) Find the value of $y$.
(ii) Show that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}8 \\ -10 \\ -1\end{array}\right)$. |
| (c) | Find the angle between $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$. |
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| 7 | (IB/sl/2018/May/paper1tz1/q9)
[Maximum mark: 16]
Point A has coordinates $(-4,-12,1)$ and point B has coordinates $(2,-4,-4)$.
| (a) | Show that $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}6 \\ 8 \\ -5\end{array}\right)$, |
| (b) | The line $L$ passes through $\mathrm{A}$ and $\mathrm{B}$.
(i) Find a vector equation for $L$.
(ii) Point $\mathrm{C}(k, 12,-k)$ is on $L$. Show that $k=14$. |
| (c) | (i) Find $\overrightarrow{O B} \cdot \overrightarrow{A B}$
(ii) Write down the value of angle $\mathrm{OBA}$. |
Point $\mathrm{D}$ is also on $L$ and has coordinates $(8,4,-9)$.
| (d) | Find the area of triangle $\mathrm{OCD}$. |
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| 8 | (IB/s1/2018/May/paper1tz2/q1)
[Maximum mark: 5]
Let $\overrightarrow{\mathrm{OA}}=\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)$ and $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{l}1 \\ 3 \\ 1\end{array}\right)$, where $\mathrm{O}$ is the origin. $L_{1}$ is the line that passes through $\mathrm{A}$ and $\mathrm{B}$.
| (a) | Find a vector equation for $L_{1}$. |
(b) The vector $\left(\begin{array}{l}2 \\ p \\ 0\end{array}\right)$ is perpendicular to $\overrightarrow{\mathrm{AB}}$. Find the value of $p$. |
| 9 | (IB/s1/2017/November/Paper1/q9)
[Maximum mark: 15]
A line $L$ passes through points $\mathrm{A}(-3,4,2)$ and $\mathrm{B}(-1,3,3)$.
| (a) | (i) Show that $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}2 \\ -1 \\ 1\end{array}\right)$.
(ii) Find a vector equation for $L$. [3] |
The line $L$ also passes through the point $\mathrm{C}(3,1, p)$.
| (b) | Find the value of $p$. |
| (c) | The point $\mathrm{D}$ has coordinates $\left(q^{2}, 0, q\right)$. Given that $\overrightarrow{\mathrm{DC}}$ is perpendicular to $L$, find the possible values of $q$. [7] |
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| 10 | (IB/sl/2017/November/Paper2/q3)
[Maximum mark: 6]
Let $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{l}4 \\ 1 \\ 2\end{array}\right)$
| (a) | Find $|\overrightarrow{\mathrm{AB}}|$. [2] |
| (b) | Let $\overrightarrow{A C}=\left(\begin{array}{l}3 \\ 0 \\ 0\end{array}\right)$. Find BẢC. |
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| 11 | (IB/sl/2017/May/paper $1 \mathrm{tz} 1 / \mathrm{q} 8$ )
[Maximum mark: 17]
A line $L_{1}$ passes through the points $\mathrm{A}(0,1,8)$ and $\mathrm{B}(3,5,2)$.
| (a) | (i) Find $\overrightarrow{\mathrm{AB}}$.
(ii) Hence, write down a vector equation for $L_{1}$. |
| (b) | A second line $L_{2}$, has equation $r=\left(\begin{array}{c}1 \\ 13 \\ -14\end{array}\right)+s\left(\begin{array}{l}p \\ 0 \\ 1\end{array}\right)$. |
Given that $L_{1}$ and $L_{2}$ are perpendicular, show that $p=2$. [3]
| (c) | The lines $L_{1}$ and $L_{2}$ intersect at $\mathrm{C}(9,13, z)$. Find $z$. |
| (d) | (i) Find a unit vector in the direction of $L_{2} .$
(ii) Hence or otherwise, find one point on $L_{2}$ which is $\sqrt{5}$ units from $\mathrm{C}$. |
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| 12 | (IB/s1/2017/May/paper1tz2/q9)
[Maximum mark: 16]
Note: In this question, distance is in metres and time is in seconds.
Two particles $P_{1}$ and $P_{2}$ start moving from a point $\mathrm{A}$ at the same time, along different straight lines.
After $t$ seconds, the position of $P_{1}$ is given by $r=\left(\begin{array}{c}4 \\ -1 \\ 3\end{array}\right)+t\left(\begin{array}{c}1 \\ 2 \\ -2\end{array}\right)$.
| (a) | Find the coordinates of $\mathrm{A}$. |
Two seconds after leaving $\mathrm{A}, P_{1}$ is at point $\mathrm{B}$.
| (b) | Find
(i) $\overrightarrow{\mathrm{AB}}:$
| (ii) $|\overrightarrow{\mathrm{AB}}|$. |
Two seconds after leaving $\mathrm{A}, P_{2}$ is at point $\mathrm{C}$, where $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{l}3 \\ 0 \\ 4\end{array}\right)$,
| (c) | Find $\cos \mathrm{BAC}$. |
| (d) | Hence or otherwise, find the distance between $P_{1}$ and $P_{2}$ two seconds after they leave $\mathrm{A}$. |
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| 13 | (IB/s1/2016/November/Paper1/q4)
[Maximum mark: 7]
The position vectors of points $\mathrm{P}$ and $Q$ are $i+2 j-k$ and $7 i+3 j-4 k$ respectively.
| (a) | Find a vector equation of the line that passes through $\mathrm{P}$ and $\mathrm{Q}$. |
| (b) | The line through $\mathrm{P}$ and $\mathrm{Q}$ is perpendicular to the vector $2 i+n k$. Find the value of $n$. |
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| 14 | (IB/sl/2016/May/paper1tz2/q7)
[Maximum mark: 7 ]
Let $u=-3 i+j+k$ and $v=m j+n k$, where $m, n \in \mathbb{R}$, Given that $v$ is a unit vector perpendicular to $u$, find the possible values of $m$ and of $n$.
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| 15 | (IB/s1/2016/May/paper2tz1/q10)
[Maximum mark: 16]
The points $\mathrm{A}$ and $\mathrm{B}$ lie on a line $L$, and have position vectors $\left(\begin{array}{c}-3 \\ -2 \\ 2\end{array}\right)$ and $\left(\begin{array}{c}6 \\ 4 \\ -1\end{array}\right)$ respectively.
Let $O$ be the origin. This is shown on the following diagram.
| (a) | Find $\overrightarrow{\mathrm{AB}}$. |
The point $\mathrm{C}$ also lies on $L$, such that $\overrightarrow{\mathrm{AC}}=2 \overrightarrow{\mathrm{CB}}$.
| (b) | Show that $\overrightarrow{O C}=\left(\begin{array}{l}3 \\ 2 \\ 0\end{array}\right)$ |
Let $\theta$ be the angle between $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{OC}}$.
Let $\mathrm{D}$ be a point such that $\overrightarrow{O D}=k \overrightarrow{O C}$, where $k>1$, Let $\mathrm{E}$ be a point on $L$ such that $\mathrm{CEED}$ is a right angle. This is shown on the following diagram.
| (d) | (i) Show that $|\overrightarrow{D E}|=(k-1)|\overrightarrow{O C}| \sin \theta$.
(ii) The distance from D to line $L$ is less than 3 units. Find the possible values of $k$. $[6]$ |
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| 16 | (IB/s1/2016/May/paper2tz2/q10)
[Maximum mark: 15]
Consider the points $\mathrm{A}(1,5,-7)$ and $\mathrm{B}(-9,9,-6)$.
| (a) | Find $\overrightarrow{A B}$. [2] |
Let $\mathrm{C}$ be a point such that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}6 \\ -4 \\ 0\end{array}\right)$
| (b) | Find the coordinates of $\mathrm{C}$. [2] |
The line $L$ passes through $\mathrm{B}$ and is parallel to (AC).
| (c) | Write down a vector equation for $L$. |
| (d) | Given that $|\overrightarrow{\mathrm{AB}}|=k|\overrightarrow{\mathrm{AC}}|$, find $k$. |
| (e) | The point $\mathrm{D}$ lies on $L$ such that $|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BD}}|$. Find the possible coordinates of $\mathrm{D}$. $[6]$ |
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| 17 | (IB/sl/2019/May/paper1tz1/q6)
[Maximum mark: 7]
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 $w$.
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| 18 | (IB/s1/2019/November/Paper1/q9)
[Maximum mark: 17]
The points $\mathrm{A}$ and $\mathrm{B}$ have position vectors $\left(\begin{array}{c}-2 \\ 4 \\ -4\end{array}\right)$ and $\left(\begin{array}{l}6 \\ 8 \\ 0\end{array}\right)$ respectively.
Point $\mathrm{C}$ has position vector $\left(\begin{array}{c}-1 \\ k \\ 0\end{array}\right)$. Let $\mathrm{O}$ be the origin.
| (a) | Find, in terms of $k$. |
(i) $\overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OC}}$;
(ii) $\overrightarrow{\mathrm{OB}} \cdot \overrightarrow{\mathrm{OC}}$.
| (b) | Given that $\mathrm{AOC}=\mathrm{BOC}$, show that $k=7$. |
| (c) | Calculate the area of triangle $\mathrm{AOC}$. |
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| 19 | (IB/s1/2018/May/paper1tz1/q6)
[Maximum mark: 6]
Six equilateral triangles, each with side length $3 \mathrm{~cm}$, 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 $p\bullet (p+q+r)$
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| 20 | (IB/s1/2018/May/paper2tz2/q8)
[Maximum mark: 13]
Two points $\mathrm{P}$ and $\mathrm{Q}$ have coordinates $(3,2,5)$ and $(7,4,9)$ respectively.
| (a) | (i) Find $\overrightarrow{P Q}$.
(ii) Find $|\overrightarrow{\mathrm{PQ}}|$,
Let $\overrightarrow{P R}=6 i-j+3 k$ |
| (b) | Find the angle between $\mathrm{PQ}$ and $\mathrm{PR}$. |
| (c) | Find the area of triangle PQR. |
| (d) | Hence or otherwise find the shortest distance from $\mathrm{R}$ to the line through $\mathrm{P}$ and $\mathrm{Q}$. |
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| 21 | (IB/s1/2017/May/paper2tz1/q2)
[Maximum mark: 6]
Let $v=\left(\begin{array}{c}-10 \\ 2 \\ 1\end{array}\right)$ and $w=\left(\begin{array}{c}3 \\ -4 \\ 0\end{array}\right)$. Find the angle between $v$ and $w$, giving your answer correct to one decimal place.
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| 22 | (IB/s1/2016/November/Paper1/q8)
[Maximum mark: 16]
Let $\overrightarrow{O A}=\left(\begin{array}{c}-1 \\ 0 \\ 4\end{array}\right)$ and $\overrightarrow{O B}=\left(\begin{array}{l}4 \\ 1 \\ 3\end{array}\right)$
| (a) | (i) Find $\overrightarrow{A B}$.
(ii) Find $|\overrightarrow{A B}| .$ |
The point $\mathrm{C}$ is such that $\overrightarrow{\mathrm{AC}}=\left(\begin{array}{c}-1 \\ 1 \\ -1\end{array}\right)$
| (b) | Show that the coordinates of $C$ are $(-2,1,3)$. |
The following diagram shows triangle $\mathrm{ABC}$. Let $\mathrm{D}$ be a point on [BC]. with acute angle $\mathrm{ADC}=\theta$
| (c) | Write down an expression in tems of $\theta$ for
(i) angle ADB:
(ii) area of triangle ABD. |
| (d) | Given that $\frac{\text { area } \Delta \mathrm{ABD}}{\text { area } \Delta \mathrm{ACD}}=3$, show that $\frac{\mathrm{BD}}{\mathrm{BC}}=\frac{3}{4}$. |
| (e) | Hence or otherwise, find the coordinates of point D. $[4]$ |
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| 23 | (IB/sl/2015/May/paper1tz1/q8)
[Maximum mark: 16]
A line $L$ passes through points $\mathrm{A}(-2,4,3)$ and $\mathrm{B}(-1,3,1)$.
| (a) | (i) Show that $\overrightarrow{A B}=\left(\begin{array}{c}1 \\ -1 \\ -2\end{array}\right)$.
(ii) Find $|\overrightarrow{\mathrm{AB}}|$. |
| (b) | Find a vector equation for $L$. [2] |
The following diagram shows the line $L$ and the origin $O$. The point $\mathrm{C}$ also lies on $L$.
Point C has position vector $\left(\begin{array}{c}0 \\ y \\ -1\end{array}\right)$
| (d) | (i) $F$ ind $\overrightarrow{\mathrm{OC}} \cdot \overrightarrow{\mathrm{AB}}$.
(ii) Hence, write down the size of the angle between $O C$ and $L$. |
| (e) | Hence or otherwise, find the area of triangle $\mathrm{OAB}$. $[4]$ |
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| 24 | (IB/sl/2015/May/paper1tz2/q9)
[Maximum mark: 15]
Let $\mathrm{P}$ and $\mathrm{Q}$ have coordinates $(1,0,2)$ and $(-11,8, \mathrm{~m})$ respectively.
| (a) | Express $\overrightarrow{\mathrm{PQ}}$ in terms of $m$. [2] |
Let $a$ and $b$ be perpendicular vectors, where $a=\left(\begin{array}{l}1 \\ 1 \\ n\end{array}\right)$ and $b=\left(\begin{array}{c}-3 \\ 2 \\ 1\end{array}\right)$.
| (c) | Given that $\overrightarrow{\mathrm{PQ}}$ is parallel to $b$.
(i) express $\overrightarrow{\mathrm{PQ}}$ in terms of $b$ :
(ii) hence find $m$. [5] |
In part (d), distance is in metres, time is in seconds.
| (d) | A particle moves along a straight line through $Q$ so that its position is given by $r=c+t a$.
(i) Write down a possible vector $c$.
(ii) Find the speed of the particle. $[4]$ |
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| 25 | (IB/sl/2015/May/paper2tz2/q2)
[Maximum mark: 7]
Let $u=6 \dot{i}+3 j+6 k$ and $v=2 i+2 j+k$.
| (a) | Find
(i) $u \cdot v$;
(ii) $|\boldsymbol{u}|$;
(iii) $|v|$. [5] |
| (b) | Find the angle between $u$ and $v$. [2] |
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| 26 | (IB/sl/2015/November/Paper1/q2)
[Maximum mark: 7 ]
The following diagram shows the parallelogram ABCD.
Let $\overrightarrow{\mathrm{AB}}=p$ and $\overrightarrow{\mathrm{AC}}=q$. Find each of the following vectors in terms of $p$ andior $q$.
| (a) | $\overrightarrow{\mathrm{CB}}$ |
| (c) | $\overrightarrow{\mathrm{DB}}$ [3] |
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| 27 | (IB/sl/2015/November/Paper1/q9)
[Maximum mark: 15]
A line $L_{1}$ passes through the points $\mathrm{A}(0,-3,1)$ and $\mathrm{B}(-2,5,3)$.
| (a) | (i) Show that $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}-2 \\ 8 \\ 2\end{array}\right)$,
(ii) Write down a vector equation for $L_{1}$. [3] |
A line $L_{2}$ has equation $r=\left(\begin{array}{c}-1 \\ 7 \\ -4\end{array}\right)+s\left(\begin{array}{c}0 \\ 1 \\ -1\end{array}\right)$, The lines $L_{1}$ and $L_{2}$ intersect at a point $C$.
| (b) | Show that the coordinates of $\mathrm{C}$ are $(-1,1,2)$. |
| (c) | A point D lies on line $L_{2}$ so that $|\overrightarrow{C D}|=\sqrt{18}$ and $\overrightarrow{C A} \cdot \overrightarrow{C D}=-9$. Find $\mathrm{ACD}$. $[7]$ |
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Answer (Vector)
1$p=-4$ or $p=3$
2(a) $k=-2$ (b) $c=\left(\begin{array}{c}6 \\ -2\end{array}\right)$
3(a) 16 (b) $r=\left(\begin{array}{c}1 \\ 2\\ 3\end{array}\right)+t\left(\begin{array}{c}6 \\ 0 \\ 2\end{array}\right)$
4(a) $ p=9 $(b) $ p=-1 $
5$(-1.29,2.64,8.07)$
6(a)(i) $\overrightarrow{A B}=\left(\begin{array}{c}11 \\ -5 \\ 3\end{array}\right)$ (ii) $12.4$ (b) (i) $y=-6$ (ii) Show (c) $.566,32.4$ (d) $42.9$
7(a) Show (b) (i) $r=\left(\begin{array}{c}-4 \\ -12 \\ 1\end{array}\right)+t\left(\begin{array}{c}6 \\ 8 \\ -5\end{array}\right)$ (ii) Show (c) (i) 0 (c) (i) 0 (ii) 90 (d) $15 \sqrt{5}$
8(a) $\quad r=\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)+t\left(\begin{array}{l}1 \\ 3 \\ 1\end{array}\right)$ (b) $p=-\frac{2}{3}$
9(a)(i) Show (ii) $r=\left(\begin{array}{c}-3 \\ 4 \\ 2\end{array}\right)+t\left(\begin{array}{c}2 \\ -1 \\ 1\end{array}\right)$ (b) $p=5$ (c) $q=-\frac{5}{2}, 2$
10(a) $|\overrightarrow{A B}|=\sqrt{21}$ (b) $B \hat{A} C=0.510(29.2)$
11(a)(i) $\overrightarrow{A B}=\left(\begin{array}{c}3 \\ 4 \\ -6\end{array}\right)$ (ii) $r=\left(\begin{array}{l}0 \\ 1 \\ 8\end{array}\right)+t\left(\begin{array}{c}3 \\ 4 \\ -6\end{array}\right)$ (b) Show (c) $z=-10$ (d) (i) $\frac{1}{\sqrt{5}}\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right)$ (ii) $(11,13,-9),(7,13,-11)$
12(a) $A=(4,-1,3)$ (b) (i) $\overrightarrow{A B}=\left(\begin{array}{c}2 \\ 4 \\ -4\end{array}\right)$ (ii) 6 (c) $\cos \widehat{B A C}=-\frac{1}{3}$ (d) 9
13(a) $r=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)+t\left(\begin{array}{c}-6 \\ -1 \\ 3\end{array}\right)$ (b) $n=4$
14(a) $m=\frac{1}{\sqrt{2}}, n=-\frac{1}{\sqrt{2}}$ or $m=-\frac{1}{\sqrt{2}}, n=\frac{1}{\sqrt{2}}$
15(a) $\overrightarrow{A B}=\left(\begin{array}{c}9 \\ 6 \\ -3\end{array}\right)$ (b) Show (b) Show (c) $\theta=0.271$ (d)(i) Show (ii) $1<k<4.11$
16(a) $\overrightarrow{A B}=\left(\begin{array}{c}-10 \\ 4 \\ 1\end{array}\right)$ (b) $C(7,1,-7)$ (c) $r=\left(\begin{array}{c}-9 \\ 9 \\ -6\end{array}\right)+t\left(\begin{array}{c}6 \\ -4 \\ 0\end{array}\right)$ (d) $\frac{3}{2}$ (e) $(0,3,-6),(-18,15,-6)$
17$\sqrt 7$
18(a)(i) $4k+2$ (ii) $8k-6$ (b) Show (c) 15
199
20(a)(i) $\quad \overrightarrow{P Q}=4 \hat{i}+2 \hat{j}+4 \hat{k}$ (ii) 6 (b) $0.582$ (c) $11.2$ (d) $3.73$
21$\theta=2.4 ; 137.9^{\circ}$
22(a)(i) $\overline{A B}=\left(\begin{array}{r}5 \\ 1 \\ -1\end{array}\right)$ (ii) $3 \sqrt{3}$ (b) Show (c) (i) $\pi-\theta$ (ii)$\frac{1}{2}|\overrightarrow{D A}||\overrightarrow{D B}| \sin (\pi-\theta)$ (d) Show (e) $\left(-\frac{1}{2}, 1,3\right)$
23](a)(i) Show (ii) $\sqrt{6}$ (b) $r=\left(\begin{array}{c}-2 \\ 4 \\ 3\end{array}\right)+t\left(\begin{array}{r}1 \\ -1 \\ -2\end{array}\right)$ (c) Show (d) (i) 0 (ii) $\frac{\pi}{2}$ (e) $\frac{\sqrt{30}}{2}$
24](a) $\overrightarrow{P Q}=\left(\begin{array}{c}-12 \\ 8 \\ m-2\end{array}\right)$ (b) $n=1$ (c) (i) $\overrightarrow{P Q}=4 \vec{b}$ (ii) $m=6$ (d) (i) $\left(\begin{array}{c}-11 \\ 8 \\ 6\end{array}\right)$ (ii) $\sqrt{3}$
25](a) (i) $\vec{u} \cdot \vec{v}=24$ (ii) $|\vec{u}|=9$ (iii) $|\vec{v}|=3$ (b) $0.476\left(27.3^{\circ}\right)$
26](a) $\quad \overrightarrow{C B}=-\vec{q}+\vec{p}$ (b) $\overrightarrow{C D}=-\vec{p}$ (c) $\overrightarrow{D B}=2 \vec{p}-\vec{q}$
27](a)(i) show (ii) $r=\left(\begin{array}{c}0 \\ -3 \\ 1\end{array}\right)+t\left(\begin{array}{c}-2 \\ 8 \\ 2\end{array}\right)$ (b) Show (c) $A \hat{C} D=\frac{2 \pi}{3}\left(\right.$ or $\left.120^{\circ}\right)$
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