Position Vector

The position vector of the point $P(x, y, z)$ from $O(0,0,0)$ is written as $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$ and $\mathrm{OP}=|\overrightarrow{\mathrm{r}}|=\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}}$

Direction cosine and direction ratios of a line

Direction cosine of a line has same meaning as d. $\mathrm{c}^{\prime} \mathrm{s}$ of a vector. If $\ell, \mathrm{m}, \mathrm{n}$ are the d. $\mathrm{c}^{\prime} \mathrm{s}$ of $\overrightarrow{\mathrm{r}}$ along OX, OY \& OZ then $\mathrm{x}=\ell \mathrm{r}, \mathrm{y}=\mathrm{mr}$ and $\mathrm{z}=\mathrm{nr}$ and $\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}=1$

Any three numbers $\mathrm{a}, \mathrm{b}, \mathrm{c}$ proportional to $\mathrm{d} . \mathrm{c}^{\prime} \mathrm{s}$ are direction ratios

i.e. $\dfrac{\ell}{\mathrm{a}}=\dfrac{\mathrm{m}}{\mathrm{b}}=\dfrac{\mathrm{n}}{\mathrm{c}}= \pm \dfrac{1}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}}$, same sign (either $+\mathrm{ve}$ or $-\mathrm{ve}$ ) should be taken throughout. Direction ratios of a line joining $\left(\mathrm{x} _{1}, \mathrm{y} _{1}, \mathrm{z} _{1}\right) \&\left(\mathrm{x} _{2}, \mathrm{y} _{2}, \mathrm{z} _{2}\right)$ are proportional to $\mathrm{x} _{2}-\mathrm{x} _{1}, \mathrm{y} _{2}-\mathrm{y} _{1}, \mathrm{z} _{2}-\mathrm{z} _{1}$. If $a _{1}, b _{1}, c _{1} \& a _{2}, b _{2}, c _{2}$ be the d. $c^{\prime} s$ of two lines, then the acute angle $\theta$ between them is given by $\cos \theta=\dfrac{\left|\mathrm{a} _{1} \mathrm{a} _{2}+\mathrm{b} _{1} \mathrm{~b} _{2}+\mathrm{c} _{1} \mathrm{c} _{2}\right|}{\sqrt{\mathrm{a} _{1}{ }^{2}+\mathrm{b} _{1}{ }^{2}+\mathrm{c} _{1}{ }^{2}} \sqrt{\mathrm{a} _{2}{ }^{2}+\mathrm{b} _{2}{ }^{2}+\mathrm{c} _{2}{ }^{2}}}$

Straight line in space

Equation of straight line is space is given by

PLANE

Equation of plane is given by

Equation of the plane passing through the line of intersection of $\mathrm{a} _{1} \mathrm{x}+\mathrm{b} _{1} \mathrm{y}+\mathrm{c} _{1} \mathrm{z}+\mathrm{d} _{1}=0$ and $\mathrm{a} _{2} \mathrm{x}+\mathrm{b} _{2} \mathrm{y}+\mathrm{c} _{2} \mathrm{z}+\mathrm{d} _{2}=0$ is $\left(\mathrm{a} _{1} \mathrm{x}+\mathrm{b} _{1} \mathrm{y}+\mathrm{c} _{1} \mathrm{z}+\mathrm{d} _{1}\right)+\lambda\left(\mathrm{a} _{2} \mathrm{x}+\mathrm{b} _{2} \mathrm{y}+\mathrm{c} _{2} \mathrm{z}+\mathrm{d} _{2}\right)=0$

Plane parallel to $\mathrm{ax}+\mathrm{by}+\mathrm{cz}+\mathrm{d}=0$ is $\mathrm{ax}+\mathrm{by}+\mathrm{cz}+\lambda=0$

Angle of intersection

Angle between

(i) two lines $\dfrac{x-x _{1}}{a _{1}}=\dfrac{y-y _{1}}{b _{1}}=\dfrac{z-z _{1}}{c _{1}}$ & $\dfrac{x-x _{2}}{a _{2}}=\dfrac{y-y _{2}}{b _{2}}=\dfrac{z-z _{2}}{c _{2}}$ is given by $\cos \theta=\dfrac{a _{1} a _{2}+b _{1} b _{2}+c _{1} c _{2}}{\sqrt{a _{1}{ }^{2}+b _{1}{ }^{2}+c _{1}{ }^{2}} \sqrt{a _{2}{ }^{2}+b _{2}{ }^{2}+c _{2}{ }^{2}}}$

(ii) two planes $\mathrm{a} _{1} \mathrm{x}+\mathrm{b} _{1} \mathrm{y}+\mathrm{c} _{1} \mathrm{z}+\mathrm{d} _{1}=0$ & $\mathrm{a} _{2} \mathrm{x}+\mathrm{b} _{2} \mathrm{y}+\mathrm{c} _{2} \mathrm{z}+\mathrm{d} _{2}=0$ given by $\cos \theta=\dfrac{a _{1} a _{2}+b _{1} b _{2}+c _{1} c _{2}}{\sqrt{a _{1}{ }^{2}+b _{1}{ }^{2}+c _{1}{ }^{2}} \sqrt{a _{2}{ }^{2}+b _{2}{ }^{2}+c _{2}{ }^{2}}}$

(iii) line $\dfrac{x-x _{1}}{a _{1}}=\dfrac{y-y _{1}}{b _{1}}=\dfrac{z-z _{1}}{z _{1}}$ & plane $a _{2} x+b _{2} y+c _{2} z+d _{2}=0$ is

given by $\sin \theta=\dfrac{a _{1} a _{2}+b _{1} b _{2}+c _{1} c _{2}}{\sqrt{a _{1}{ }^{2}+b _{1}{ }^{2}+c _{1}{ }^{2}} \sqrt{a _{2}{ }^{2}+b _{2}{ }^{2}+c _{2}{ }^{2}}}$

Distance of a point $\left(\mathrm{x} _{1}, \mathrm{y} _{1}, \mathrm{z} _{1}\right)$ from

$\mathrm{ax}+\mathrm{by}+\mathrm{cz}+\mathrm{d}=0 \text { is }\left|\dfrac{\mathrm{ax} _{1}+\mathrm{by} _{1}+\mathrm{cz} \mathrm{z} _{1}+\mathrm{d}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}}\right|$

Distance of a point $P\left(x _{2}, y _{2}, z _{2}\right)$ from $\dfrac{x-x _{1}}{a}=\dfrac{y-y _{1}}{b}=\dfrac{z-z _{1}}{c}$ is given by

$d=\dfrac{|\overrightarrow{P A} \times \vec{b}|}{|\vec{b}|} \text { where } A \text { is }\left(x _{1}, y _{1}, z _{1}\right)$ & $\vec{b}=a \hat{i}+b \hat{j}+c \hat{k}$

• Distance between parallel planes

$ax + by +c z+d _{1}=0$ & $ ax + by +c z+d _{2}=0 $ is given by $\dfrac{\left|d _{1}-d _{2}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$

• Shortest distance between two skew lines $\vec{r}=\vec{a} _{1}+\lambda \vec{b} _{1}, \vec{r}=\vec{a} _{2}+\mu \vec{b} _{2}$, is given by

S.D $=\left|\dfrac{\left(\overrightarrow{\mathrm{a}} _{2}-\overrightarrow{\mathrm{a}} _{1}\right) \overrightarrow{\mathrm{b}} _{1} \times \overrightarrow{\mathrm{b}} _{2}}{\left|\overrightarrow{\mathrm{b}} _{1} \times \overrightarrow{\mathrm{b}} _{2}\right|}\right|$

• Distance between two parallel lines $\vec{r}=\vec{a} _{1}+\lambda \vec{b}$ & $\vec{r}=\vec{a} _{2}+\mu \vec{b}$ is given by

$\overrightarrow{\mathrm{r}}=\dfrac{\left|\left(\overrightarrow{\mathrm{a}} _{2}-\overrightarrow{\mathrm{a}} _{1}\right) \times \overrightarrow{\mathrm{b}}\right|}{|\overrightarrow{\mathrm{b}}|}$

• Image $(\alpha, \beta, \gamma)$ of $\left(x _{2}, y _{2}, z _{2}\right)$ in a straight line $\dfrac{x-x _{1}}{a}=\dfrac{y-y _{1}}{b}=\dfrac{z-z _{1}}{c}$ is given by,

$\begin{aligned} & \alpha=2\left(\mathrm{a} \lambda+\mathrm{x} _{1}\right)-\mathrm{x} _{2}, \beta=2\left(\mathrm{~b} \lambda+\mathrm{y} _{1}\right)-\mathrm{y} _{2} ; \gamma=2\left(\mathrm{c} \lambda+\mathrm{z} _{1}\right)-\mathrm{z} _{2} \text { where } \\ & \lambda=\dfrac{\mathrm{a}\left(\mathrm{x} _{2}-\mathrm{x} _{1}\right)+\mathrm{b}\left(\mathrm{y} _{2}-\mathrm{y} _{1}\right)+\mathrm{c}\left(\mathrm{z} _{2}-\mathrm{z} _{1}\right)}{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}} \end{aligned}$

• Image of $\left(\mathrm{x} _{1} \mathrm{y} _{1} \mathrm{z} _{1}\right)$ in the plane $\mathrm{ax}+\mathrm{by}+\mathrm{cz}+\mathrm{d}=0$ is $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ where

$\dfrac{\mathrm{x}-\mathrm{x} _{1}}{\mathrm{a}}=\dfrac{\mathrm{y}-\mathrm{y} _{1}}{\mathrm{~b}}=\dfrac{\mathrm{z}-\mathrm{z} _{1}}{\mathrm{c}}=\dfrac{-2\left|\mathrm{ax} _{1}+\mathrm{by} _{1}+\mathrm{cz} _{1}+\mathrm{d}\right|}{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}$

Bisecting planes

Equation of the planes bisecting the angle between the planes $a _{1} x+b _{1} y+c _{1} z+d _{1}=0$ and $a _{2} x+b _{2} y+c _{2} z+d _{2}=0$ is $\dfrac{a _{1} x+b _{1} y+c _{1} z+d _{1}}{\sqrt{a _{1}{ }^{2}+b _{1}{ }^{2}+c _{1}{ }^{2}}}= \pm \dfrac{a _{2} x+b _{2} y+c _{2} z+d _{2}}{\sqrt{a _{2}{ }^{2}+b _{2}{ }^{2}+c _{2}{ }^{2}}}$

A program to give wings to girl students

Bisector of acute and obtuse angles between two planes

Line of Greatest slope

$\mathrm{AB}$ is the line of intersection of plane $\pi _{1}$ & $\pi _{2}$. Line of greatest slope on a given plane, drawn through $\mathrm{P}$ and perpendicular to the line of intersection of the given plane with any horizontal plane.

System of planes (Relationship between planes)

In $3 \mathrm{D}, 3$ planes are given by

$\pi _{1} \equiv \mathrm{a} _{1} \mathrm{x}+\mathrm{b} _{1} \mathrm{y}+\mathrm{c} _{1} \mathrm{z}+\mathrm{d} _{1}=0$

$\pi _{2} \equiv \mathrm{a} _{2} \mathrm{x}+\mathrm{b} _{2} \mathrm{y}+\mathrm{c} _{2} \mathrm{z}+\mathrm{d} _{2}=0$

$\pi _{2} \equiv \mathrm{a} _{3} \mathrm{x}+\mathrm{b} _{3} \mathrm{y}+\mathrm{c} _{3} \mathrm{z}+\mathrm{d} _{3}=0$

Let $\mathrm{r}$ & $\mathrm{~s}$ be the rank of coefficient matrix and augmented matrix

Case I

Planes intersect at a point if $r=3 \& s=3$

Case II

Each plane cut the other two in a line if $\mathrm{r}=2$ $\& s=3$. Planes form a prismatic surface

2. parallel planes and the other cuts each in aline.

Two rows of the coefficient matrix are proportional

$\dfrac{\mathrm{a}}{\mathrm{a}^{\ast}}=\dfrac{\mathrm{b}}{\mathrm{b}^{\ast}}=\dfrac{\mathrm{c}}{\mathrm{c}^{\ast}} \neq \dfrac{\mathrm{d}}{\mathrm{d}^{\ast}}$

Case III

3planes interesting in a line if $\mathrm{r}=2 \& \mathrm{~s}=2$

2 coincident planes \& the other intersecting them in a line if $\mathrm{r}=2 & \mathrm{~s}=2$ 2 rows of the augmented matrix are

proportional $\dfrac{\mathrm{a}}{\mathrm{a}^{\ast}}=\dfrac{\mathrm{b}}{\mathrm{b}^{\ast}}=\dfrac{\mathrm{c}}{\mathrm{c}^{\ast}}=\dfrac{\mathrm{d}}{\mathrm{d}^{\ast}}$

Case IV

Three parallel planes if $\mathrm{r}=1 \mathrm{~s}=2$

Two coincident planes \& the other parallel if $\mathrm{r}=1, \mathrm{~s}=2$ Two rows of the argument matrix are proportional

$\dfrac{\mathrm{a}}{\mathrm{a}^{\ast}}=\dfrac{\mathrm{b}}{\mathrm{b}^{\ast}}=\dfrac{\mathrm{c}}{\mathrm{c}^{\ast}}=\dfrac{\mathrm{d}}{\mathrm{d}^{\ast}}$

Case V

3 coincident planes $\mathrm{r}=1 \& \mathrm{~s}=1$

SOLVED EXAMPLES

1. Find the image of $(1,3,4)$ in the plane $x+2 y-z+3=0$ is

(a) $(1,1,-6)$

(b) $(-1,-1,6)$

(c) $(-1,1-6)$

(d) None of these

2. The distance of $\mathrm{P}(0,2,3)$ to the line $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-1 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$

$\mathrm{A}(3,1,-1), \mathrm{P}(0,2,3)$

3. $\mathrm{ABCD}$ is a square of unit side. It is folded along the diagonal $\mathrm{AC}$ so that the plane $\mathrm{ABC}$ is perpendicular to the plane ACD. The shortest distance between the lines AB \& CD is

(a) $\sqrt{\dfrac{3}{2}}$

(b) $\dfrac{3}{2}$

(c) $\dfrac{2}{3}$

(d) $\sqrt{\dfrac{2}{3}}$

Show Answer

Solution

Let $\mathrm{A}(1,0,0) \mathrm{B}(1,1,0), \mathrm{C}(0,1,0), \mathrm{D}(0,0,0)$

Equation of $\mathrm{CD}$ is

$$ \begin{equation*} \overrightarrow{\mathrm{r}}=\overrightarrow{0}+\lambda \hat{\mathrm{j}} \tag{1} \end{equation*} $$

New position of $\mathrm{B}$ is $\left(\dfrac{1}{2}, \dfrac{1}{2}, \dfrac{1}{\sqrt{2}}\right)$

Equation of $\mathrm{AB}$ is

$\overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\mu\left(-\dfrac{1}{2} \hat{\mathrm{i}}+\dfrac{1}{2} \hat{\mathrm{j}}+\dfrac{1}{\sqrt{2}} \hat{\mathrm{k}}\right) \quad \overrightarrow{\mathrm{a}} _{2}-\overrightarrow{\mathrm{a}} _{1}=\hat{\mathrm{i}}$

$\overrightarrow{\mathrm{b}} _{1} \times \overrightarrow{\mathrm{b}} _{2}=\hat{\mathrm{j}} \times\left(-\dfrac{1}{2} \hat{\mathrm{i}}+\dfrac{1}{2} \hat{\mathrm{j}}+\dfrac{1}{\sqrt{2}} \hat{\mathrm{k}}\right)=\dfrac{1}{2} \hat{\mathrm{k}}+\dfrac{1}{2} \hat{\mathrm{i}}$

S.D. b|w AB & CD is $\left|\dfrac{\left(\vec{a} _{2}-\vec{a} _{2}\right) \cdot \vec{b} _{1} \times \vec{b} _{2}}{\left|\vec{b} _{1} \times \vec{b} _{2}\right|}\right|$

$=\dfrac{\hat{i} \cdot\left(\dfrac{1}{2} \hat{k}+\dfrac{1}{\sqrt{2}} \hat{\mathrm{i}}\right)}{\left|\dfrac{1}{2} \hat{\mathrm{k}}+\dfrac{1}{\sqrt{2}} \hat{\mathrm{i}}\right|}=\dfrac{\dfrac{1}{\sqrt{2}}}{\sqrt{\dfrac{1}{4}+\dfrac{1}{2}}}=\dfrac{1}{\sqrt{2}} \cdot \dfrac{2}{\sqrt{3}}=\sqrt{\dfrac{2}{3}}$

Answer (d)

4. For what value of $\lambda$ do the planes $x-y+z+1=0, \lambda x+3 y+2 z-3=0,3 x+\lambda y+z-2=0$ form a triangular prism?

(a) 4

(b) -3

(c) 2

(d) $\dfrac{16}{3}$

5. The plane $\ell \mathrm{x}+\mathrm{my}=0$ is rotated about its line of intersection with the XOY plane through an angle $\alpha$, then the equation of the plane is $\ell \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ where $\mathrm{n}$ is

(a) $\pm \sqrt{\left(\ell^{2}+m^{2}\right)} \cos \alpha$

(b) $\pm \sqrt{\left(\ell^{2}-m^{2}\right)} \sin \alpha$

(c) $\pm \sqrt{\left(\ell^{2}+m^{2}\right)} \tan \alpha$

(d) $\pm \sqrt{\left(\ell^{2}-m^{2}\right)} \sec \alpha$

6. State the relationship between the planes

$\pi _{1} \equiv 2 \mathrm{x}-3 \mathrm{y}+4 \mathrm{z}-1=0$

$\pi _{2} \equiv \mathrm{x}-\mathrm{y}-\mathrm{z}+1=0$

$\pi _{3} \equiv-\mathrm{x}+2 \mathrm{y}-\mathrm{z}+2=0$

Exercise

1. If $z$-axis be vertical, then the equation of the line of greatest slope through the point $(2,-1,0)$ on the plane $2 x+3 y-4 z=1$ is

(a) $\dfrac{x-2}{2}=\dfrac{y+1}{-1}=\dfrac{z}{0}$

(b) $\dfrac{x-2}{3}=\dfrac{y+1}{4}=\dfrac{z}{5}$

(c) $\dfrac{\mathrm{z}-6}{-6}=\dfrac{\mathrm{y}+1}{4}=\dfrac{\mathrm{z}}{12}$

(d) $\dfrac{x-2}{8}=\dfrac{y+1}{12}=\dfrac{z}{13}$

2. Through a point $\mathrm{P}(\mathrm{f}, \mathrm{g}, \mathrm{h})$ a plane is drawn at right angles to $\mathrm{OP}$, to meet the axes in $\mathrm{A}, \mathrm{B}, \mathrm{C}$. If $\mathrm{O}$ is $(0,0,0)$ the centroid of $\triangle \mathrm{ABC}$ is

(a) $\left(\dfrac{\mathrm{f}}{3 \mathrm{r}}, \dfrac{\mathrm{g}}{3 \mathrm{r}}, \dfrac{\mathrm{h}}{3 \mathrm{r}}\right)$

(b) $\left(\dfrac{\mathrm{r}^{2}}{3 \mathrm{f}^{2}}, \dfrac{\mathrm{r}^{2}}{3 \mathrm{~g}^{2}}, \dfrac{\mathrm{r}^{2}}{3 \mathrm{~h}^{2}}\right)$

(c) $\left(\dfrac{\mathrm{r}^{2}}{3 \mathrm{f}}, \dfrac{\mathrm{r}^{2}}{3 \mathrm{~g}}, \dfrac{\mathrm{r}^{2}}{3 \mathrm{~h}}\right)$

(d) $\left(\dfrac{\mathrm{r}^{2}}{3 \mathrm{r}^{2}}, \dfrac{\mathrm{g}^{2}}{3 \mathrm{r}^{2}}, \dfrac{\mathrm{h}^{2}}{3 \mathrm{r}^{2}}\right)$

3. If a line with d. $\mathrm{r}^{\prime} \mathrm{s} 2: 2: 1$ intersects the line $\dfrac{\mathrm{x}-7}{3}=\dfrac{\mathrm{y}-5}{2}=\dfrac{\mathrm{z}-3}{1}$ and $\dfrac{\mathrm{x}-1}{2}=\dfrac{\mathrm{y}+1}{4}=$ $\dfrac{\mathrm{z}+1}{3}$ at $\mathrm{A} \& \mathrm{~B}$, then $\mathrm{AB}=$

(a) $\sqrt{2}$

(b) 2

(c) $\sqrt{3}$

(d) 3

4. A mirror and a source of light are situated at the origin and a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the d. $\mathrm{r}^{\prime} \mathrm{s}$ of the normal to the plane of mirror are $1,-1,1$ then the dc’s for the reflectd ray are

(a) $\dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3}$

(b) $\dfrac{1}{3}, \dfrac{-2}{3}, \dfrac{-2}{3}$

(c) $\dfrac{-1}{3}, \dfrac{-2}{3}, \dfrac{-2}{3}$

(d) $\dfrac{-1}{3}, \dfrac{-2}{3}, \dfrac{2}{3}$

5. Let $\mathrm{A}(\vec{a})$ and $\mathrm{B}(\vec{b})$ be points on two skew lines $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{b}}+\mu \overrightarrow{\mathrm{q}}$ and the shortest distance between the skew lines is 1 , where $\vec{p}$ and $\vec{q}$ are unit vectors forming adjacent sides of a parallelogram enclosing an area of $\dfrac{1}{2}$ sq.units. If an angle between $\mathrm{AB}$ and the line of shortest distance is $60^{\circ}$, then $\mathrm{AB}=$

(a) $1 / 2$

(b) 2

(c) 1

(d) None of these

6.* $\mathrm{A}(-1,2,-3), \mathrm{B}(5,0,-6)$ and $\mathrm{c}(0,4,-1)$ are the vertices of the $\triangle \mathrm{ABC}$ then

(a) The d. $\mathrm{r}^{\prime} \mathrm{s}$ of the internal bisector of $\angle \mathrm{BAC}$ are $(25,8,5)$

(b) The d.r’s of the internal bisector of $\angle \mathrm{ABC}$ are $6 \sqrt{66}-35,-2 \sqrt{66}+28,35-3 \sqrt{66}$

(c) The d.r’s of the internal bisector of $\angle \mathrm{BCA}$ are $\sqrt{66}-15,2 \sqrt{66}+12,2 \sqrt{66}+15$

(d) The d. r’ $s$ of the internal bisector of $\angle \mathrm{BCA}$ are 24,6,3

7.* Consider the line $\mathrm{x}=\mathrm{y}=\mathrm{z}$ and the line $2 \mathrm{x}+\mathrm{y}+\mathrm{z}-1=0=3 \mathrm{x}+\mathrm{y}+2 \mathrm{z}-2$ then

(a) The shortest distance between the two lines is $\dfrac{1}{\sqrt{2}}$

(b) The shortest distance between the two lines is $\sqrt{2}$

(c) plane containing $2^{\text {nd }}$ line parallel to $1^{\text {st }}$ line is $y-z+1=0$

(d) The shortest between the two lines is $\dfrac{\sqrt{3}}{2}$

8. Read the passage and answer the questions that follow.

Given that $\vec{a}=6 \hat{i}+7 \hat{j}+7 \hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}-7 \hat{k} . P(1,2,3)$

(i) The position vector of $L$, the foot of the $\perp r$ from $P$ on the line $\vec{r}=\vec{a}+\lambda \vec{b}$ is

(a) $6 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$

(b) $3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$

(c) $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+9 \hat{\mathrm{k}}$

(d) $9 \hat{i}+9 \hat{j}+5 \hat{k}$

(ii) The image of the point $P$ in the line $\vec{r}=\vec{a}+\lambda \vec{b}$ is

(a) $(11,12,11)$

(b) $(5,2,-7)$

(c) $(5,8,15)$

(d) $(17,16,16,7)$

(iii) If A is the point with position vector $\vec{a}$ then area of the triangle $\triangle \mathrm{PLA}$ in sq.units is equal to

(a) $3 \sqrt{6}$

(b) $\dfrac{7 \sqrt{17}}{2}$

(c) $\sqrt{17}$

(d) $\dfrac{7}{2}$

9. The position vector of the point on the line $\vec{r}=\hat{i}+\lambda(\hat{i}+\hat{j}+\hat{k})$ whose distance from $\vec{r}-(\hat{i}+\hat{j})$. $(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathrm{k}})=1$ is $\dfrac{1}{\sqrt{3}}$ units is

(a) $-\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$

(b) $\hat{j}$

(c) $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$

(d) $3 \hat{i}+2 \hat{j}+2 \hat{k}$

10. Match the following