Question 1 : Find shortest distance between the sides of parrallelogram $\overline r=2\overline i-\overline j+\lambda (2\overline i+\overline j-3\overline k)$ and $\overline r=\overline i- \overline j+2\overline k+\mu (2\overline i+\overline j-5\overline k)$<p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p>

Question 2 : The position vector of point $A$ is $(4, 2, -3)$. If $p_{1}$ is perpendicular distance of $A$ from $XY-plane$ and $p_{2}$ is perpendicular distance from Y-axis, then $p_{1} + p_{2} =$ _______.

Question 3 : A square $ABCD$ of diagonal $2a$ is folded along the diagonal $AC$ so that the planes $DAC$ and $BAC$ are at right angle. The shortest distance between $DC$ and $AB$ is

Question 4 : The vector equation $r=i-2j-k+t(6j-k)$ represents a straight line passing through the points:

Question 6 : Equation to a line parallel to the vector $2\hat{i}-\hat{j}{+}\hat{k}$ and passing through the point $\hat{i}+\hat{j}{+\hat{k}}$<br/>

Question 7 : If $L_1 = 0$ is the reflected ray, then its equation is

Question 8 : Find the shortest distance between two lines : $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$ is

Question 10 : $L$ and $M$ are two points with position vectors $2\overline { a } -\overline { b } $ and $a+2\overline { b } $ respectively. The position vector of the point $N$ which divides the line segment $LM$ in the ratio $2:1$ externally is

Question 11 : <span>The vector form of the equation of the line passing through points $(3,4, 7)$ and $(5,1,6)$ is-</span>

Question 12 : The shortest distance between the skew lines $\frac { x - 3 } { - 1 } = \frac { y - 4 } { 2 } = \frac { z + 2 } { 1 } , \frac { x - 1 } { 1 } = \frac { y + 7 } { 3 } = \frac { z + 2 } { 2 }$ is

Question 13 : The line perpendicular to the plane $2x-y+5z=4$ passing through the point $(-1,0,1)$ is ?<br/>

Question 14 : Find the vector equation of line joining the points $ (2,1,3)$ and $(-4,3,-1)$

Question 16 : The length of the perpendicular drawn from the points $(5,4,-1)$ to the line $\overline r = \widehat i + \lambda \left( {2\widehat i + 9\widehat i + 5\widehat k} \right)$ is

Question 17 : The equation of the plane passing through $(a,b,c)$ and parallel to the plane $r.(\hat{i}+\hat{j}+\hat{k})=2$ is,

Question 18 : Find vector equation for the line passing through the points $3\overline i+4\overline j-7\overline k,\overline i-\overline j+6\overline k$.<p></p><p></p><p></p><p></p><p></p><p></p>

Question 19 : If the normal of the plane makes an angles $\dfrac {\pi}{4}, \dfrac {\pi}{4}$ and $\dfrac {\pi}{2}$ with positive X-axis, Y-axis and Z-axis respectively and the length of the perpendicular line segment from origin to the plane is $\sqrt {2}$, then the equation of the plane is ________.

Question 21 : Find the shortest distance between the lines $\overline r=4\overline i-\overline j+\lambda (\overline i+2\overline j-5\overline k)$ and $\overline r=\overline i-\overline j+2\overline k+ \mu (\overline i+2\overline j-5\overline k)$<p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p>

Question 22 : The length of perpendicular from the origin to the plane which makes intercepts $\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5}$ respectively on the coordinate axes is

Question 23 : The vector equation of line passing through two points $A(x_1,y_1,z_1),B(x_2,y_2,z_2) $ is<br/>

Question 25 : If $P$ is a point on the line passing through the point $A$ with position vector $2\overline{i}+\overline{j}-3\overline{k}$ and parallel to $\overline{i}+2\overline{j}+\overline{k}$ such that $AP=2\sqrt{6}$ then the position vector of $P$ is<br/>

Question 26 : The shortest distance betwwen lines $\overline r=2\overline i-\overline j+\lambda(2\overline i+\overline j-3\overline k)$ and $\overline r=\overline i-\overline j+2 \overline k+\mu (2\overline i+\overline j-5\overline k)$

Question 27 : Vector equation of the line $6x - 2 = 3y + 1 = 2z - 2$ is

Question 28 : The perpendicular distance of the point $(2,4,-1)$ from the line $\dfrac{x+5}{1}=\dfrac{y+3}{4}=\dfrac{z-6}{-9}$ is

Question 29 : Perpendicular distance between the plane $ 2 x-y+2 z=1 $ and origin is

Question 31 : The vector equation of a plane passing through a point whose. P.V. is $\overrightarrow a$ and perpendicular to a vector $\overrightarrow n$, is

Question 34 : The shortest distance between the lines $\dfrac {x - 7}{3} = \dfrac {y + 4}{-16} = \dfrac {z - 6}{7}$ and $\dfrac {x - 10}{3} = \dfrac {y- 30}{8} = \dfrac {4 - z}{-5}$ is

Question 35 : A line is perpendicular to the plane $x+2y+2z=0$ and passes through $(0, 1, 0)$. The perpendicular distance of this line from the origin is

Question 36 : The perpendicular distance of the point $\left ( x,\, y,\, z \right )$ from the x-axis is <br>

Question 37 : The shortest distance of the points $(a, b, c)$ from the x-axis is

Question 38 : If O is the origin and $P(1, 2, -3)$ is a given point, then the equation of the plane through P and perpendicular to OP is?

Question 39 : <span>Perpendiculars AP, AQ and AR are drawn to the $x-,y-$ and $z-$axes, respectively, from the point $A\left ( 1,-1,2 \right )$. </span><span>The A.M. of $AP^2,$ $AQ^2$ and $AR^2$ is</span>

Question 40 : The shortest distance between the lines $x+\alpha =2y=12z$ and $x=y+29=6\left ( z-\alpha \right )$

Question 41 : <div>State true or false.</div>The equation of a line, which is parallel to $ 2 \hat{i}+\hat{j}+3 \hat{k} $ and which passes through the point (5,-2,4) is $ \dfrac{x-5}{2}=\dfrac{y+2}{-1}=\dfrac{z-4}{3} $

Question 42 : Two lines ${ L }_{ 1 }:x=5,\cfrac { y }{ 3-\alpha } =\cfrac { z }{ -2 } ,{ L }_{ 2 }:x=\alpha,\cfrac { y }{ -1 } =\cfrac { z }{ 2-\alpha } $ are coplanar. Then, $\alpha$ can take value $(s)$

Question 43 : A square $ABCD$ of disgonal $2a$ is folded along the diagonal $AC$ so that the planes $DAC$ and $BAC$ are at right angle. The shortest distance between $DC$ and $AB$ is

Question 44 : The distance between the lines $\displaystyle \frac {x-4}{2}=\frac {y+1}{-3}=\frac {z}{6}$ and $\displaystyle \frac {x}{-1}=\frac {y-1}{\dfrac {3}{2}}=\frac {z+1}{-3}$ is <br/>

Question 46 : The shortest distance between the lines $\displaystyle \frac{x - 3}{3} = \frac{y - 8}{-1} = \frac{z - 3}{1}$ and $\displaystyle \frac{x + 3}{-3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is

Question 47 : Vector equation of the plane $\vec{r}=\widehat{i}-\widehat{j}+\lambda (\widehat{i}+\widehat{j}+\widehat{k})+ \mu (\widehat{i}-2\widehat{j}+3\widehat{k})$ in the scalar dot product form is<br>

Question 48 : The distance of the point $B$ with position vector $i +2j +3k$ from the line passing through the point $A$ with position vector $4i + 2j + 2k$ and parallel to the vector $2i + 3j + 6k$ is<br>

Question 50 : The shortest distance between the lines $\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}$ and $\dfrac{x-2}{3} = \dfrac{y-4}{4} = \dfrac{z-5}{5}$, is

Question 51 : The shortest distance between the lines $x = y + 2 = 6z - 6$ and $x + 1 = 2y = -12z$ is

Question 52 : Shortest distance between the lines $\dfrac {x-1}{1}=\dfrac {y-1}{1}=\dfrac {z-1}{1}$ and $\dfrac {x-2}{1}=\dfrac {y-3}{1}=\dfrac {z-4}{1}$ is equal to-

Question 53 : If the foot of the perpendicular from $(0,0,0)$ to a plane is $P(1,2,2)$. Then, the equation of the plane is

Question 54 : If $\overline { a } ,\overline { b } ,\overline { c } $ are mutually perpendicular vectors having magnitudes $1,2,3$ respectively, then $\left[ \overline { a } +\overline { b } +\overline { c } \quad \overline { b } -\overline { a } -\overline { c } \right] =\quad $

Question 55 : The perpendicular distance from the point of intersection of the line $\cfrac { x+1 }{ 2 } =\cfrac { y+2 }{ 3 } =\cfrac { z }{ -1 } $ and plane $2x-y+z=0$ to the Z-axis is ____

Question 56 : The shortest distance between the $y$-axis and the line $2x+3y+5z+1=3x+4y+6z+2=0$ is $\displaystyle\frac{2}{\sqrt{k}}$, then the value of $'k'$ is

Question 57 : The length of the perpendicular drawn from $(1, 2, 3)$ to the line $\dfrac {x-6}{3}=\dfrac {y-7}{2}=\dfrac {z-7}{-2}$ is-

Question 58 : The direction ratios of a normal to the plane passing through $(0,1,1), (1,1,2)$ and $(-1,2,-2)$ are<br>

Question 59 : The vector equation of the line $\displaystyle \frac { x-2 }{ 2 } =\frac { 2y-5 }{ -3 } ,z=-1$ is $\displaystyle \overrightarrow { r } =\left( 2\hat { i } +\frac { 5 }{ 2 } \hat { j } -\hat { k } \right) +\lambda \left( 2\hat { i } -\frac { 3 }{ 2 } \hat { j } +x\hat { k } \right) $, where $x$ is equal to

Question 60 : The shortest distance between the lines$\bar {r}=3\bar {i}+5\bar {j}+7\bar {k}+\lambda(\bar {i}+2\bar {j}+\bar {k})$<span>$\bar {r}=-\bar {i}-\bar {j}+\bar {k}+\mu(7\bar {i}-6\bar {j}+\bar {kk})$ is ?<br/></span>

Question 61 : If a plane passes through the point $(1, 1, 1)$ and is perpendicular to the line $\dfrac{x-1}{3}=\dfrac{y-1}{0}=\dfrac{z-1}{4}$ then its perpendicular distance from the origin is

Question 62 : lf the equation of the plane perpendicular to the $\mathrm{z}$ -axis and passing through the point $(2, -3,4)$ is $ax+by+cz=d$ then $\displaystyle \dfrac{a+b+c}{d}=$<br/>

Question 63 : The equation of the plane passing through the lines $\frac {x-4}{1}=\frac {y-3}{1}=\frac {z-2}{2}$ and $\frac {x-3}{1}=\frac {y-2}{-4}=\frac {z}{5}$ is-

Question 65 : The curve $ y={x}^{3} + {x}^{3} -x $ has two horizontal tangents.The distance between these two horizontal lines,is:

Question 66 : The shortest distance between the skew lines<br/>$\vec {r}=(\hat{i}+2\hat {j}+3\hat{k})+t(\hat{i}+3\hat{j}+2\hat{k})$ and $\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+t(2\hat{i}+3\hat{j}+\hat{k})$ is

Question 67 : If $(2, 3, -1)$ is the foot of the perpendicular from $(4, 2, 1)$ to a plane, then the equation of that plane is $ax+by+cz=d$. Then $a+d$ is<br/>

Question 68 : The shortest distance between the skew lines $\overrightarrow{r}=3\hat{i}+8\hat{j}+3\hat{k}+\alpha(3\hat{i}-\hat{j}+\hat{k})$ and $\overrightarrow {r}=-3\hat{i}-7\hat{j}+6\hat{k}+\beta(-3\hat{i}+2\hat{j}+4\hat{k})$ is<br/>

Question 69 : Distance between two parrallel lines,<br/> $\overline r=\overline a_1+\lambda \overline b$ and <span>$\overline r=\overline a_2+\mu \overline b$, </span><span>is given by</span><p></p><p></p><p></p><p></p><p></p>

Question 70 : The shortest distance between the lines $\dfrac {x}{2} = \dfrac {y}{2} = \dfrac {z}{1}$ and $\dfrac {x + 2}{-1} = \dfrac {y - 4}{8} = \dfrac {z - 5}{4}$ lies in the interval:<br/>

Question 71 : If the shortest distance between the lines $\displaystyle \frac{x - 3}{3} = \frac{y - 8}{-1} = \frac{z - 3}{1}$ and $\displaystyle \frac{x + 3}{-3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is $\lambda \sqrt{30}$ units, then the value of $\lambda$ is .....................

Question 72 : Distance of the point $ (\alpha, \beta, \gamma) $ from $ y $ -axis is

Question 73 : A plane passes through the point $(0, -1, 0)$ and $(0, 0, 1)$ and makes an angle of $\dfrac{\pi}{4}$ with the plane $y-z=0$ then the point which satisfies the desired plane is?

Question 74 : The direction cosines of the normal to the plane $\displaystyle 5\left ( x - 2 \right ) = 3\left ( y - z \right )$ are

Question 75 : The equation of the plane passing through the point $(-1, 2, 1)$ and perpendicular to the line joining the points $(-3, 1, 2)$ and $(2, 3, 4)$ is ________.

Question 76 : If $A=(-1,2,-3)$ , $B=(-16,6,4)$, $C=(1,-1,3)$ and $D=(4,9, 7)$, then the shortest distance between the lines $\vec {AB},\ \vec {CD}$ is<br/>

Question 77 : Lines $OA,OB$ are drawn from $O$ with direction cosines proportional to $(1,-2,-1),(3,-2,3).$ Find the direction cosines of the normal to the plane $AOB$

Question 79 : Equation of the line of the shortest distance between<span>the lines $\frac { x } { 2 } = \frac { y } { - 3 } = \frac { z } { 1 }$ and $\frac { x - 2 } { 3 } = \frac { y - 1 } { - 5 } = \frac { z + 2 } { 2 }$ is</span>

Question 80 : <div>Consider the line</div>$\displaystyle {L}_{1}:\dfrac{{x}+1}{3}=\dfrac{{y}+2}{1}=\dfrac{{z}+1}{2},\ \displaystyle {L}_{2}:\dfrac{{x}-2}{1}=\dfrac{{y}+2}{2}=\dfrac{{z}-3}{3}$<div>The shortest distance between $L_1$ and $L_2$ is<br/></div>

Question 82 : Assertion: The shortest distance between the skew lines $\displaystyle \frac{x-1}{2}= \frac{x-1}{-1}= \frac{z-0}{1}$ and $\displaystyle \frac{x-2}{3}= \frac{y-1}{-5}= \frac{z+1}{2}$ is $\displaystyle \frac{10}{\sqrt{59}}$ Reason: Two lines are skew if there exists no plane passing through them

Question 83 : &#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;&#10;<p class="MsoNormal"><span>If the straight lines&#10;$\displaystyle \dfrac{x-1}{2}=\dfrac{y+1}{K}=\dfrac{z}{2}$ and $\displaystyle&#10;\dfrac{x+1}{5}=\dfrac{y+1}{2}=\dfrac{z}{K}$ are coplanar, then the plane(s)&#10;containing these two lines is(are)</span></p>&#10;&#10;

Question 84 : The shortest distance between the $z$-axis and the line $x+ y + 2z - 3 = 0 , 2x + 3y + 4z - 4 $, is

Question 85 : The, position vector of the foot of the $\perp$er drawn from origin to the plane is $\displaystyle 4\hat{i}-2\hat{j}-5\hat{k} $ then equation of the plane is<br>

Question 86 : If $(2, -1, 3)$ is the foot of the perpendicular drawn from the origin to the plane, then the equation of the plane is

Question 87 : The perpendicular distance of the point $P(1,2,3)$ from the straight line passing through the point $A(-1,4,7)$ and $B(2,8,7)$

Question 88 : The shortest distance between the lines $\vec{r}=(4\hat{i}-\hat{j})+ \lambda(\hat{i}+2\hat{j}-3\hat{k}),\lambda\epsilon R$ and $\vec{r}=(-\hat{i}-\hat{j}+2\hat k)+\mu(2\hat{i}+4\hat{j}-5\hat{k}),\mu\epsilon R$ is <br/>

Question 89 : The equation of the line parallel to $\cfrac { x-3 }{ 1 } =\cfrac { y+3 }{ 5 } =\cfrac { 2z-5 }{ 3 } $ and passing through the point $(1,3,5)$ in vector form, is:

Question 90 : A line is drawn from $P(x_1 , y_1)$ in the direction $\theta$ with the X - axis, to meet <b></b>$ax + by + c = 0$ at $Q$. Then length $PQ$ is equal to :

Question 91 : The shortest distance between the lines whose equations are $\displaystyle \vec{r}=t\left ( \hat{i}+\hat{j}+\hat{k}\right )$ and $\displaystyle \vec{r}= \hat{k}+s\left ( \hat{i}+\hat{2j}+\hat{3k}\right )$ is

Question 92 : The perpendicular distance of$\overrightarrow A $ (1,4,-2) from the segment BC where$\overrightarrow B $ (2,1,-2) and $\overrightarrow C $ (o,-5,1) is

Question 93 : If the shortest distance between the lines $\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda\begin{pmatrix}2\hat{i}+3\hat{j}+4\hat{k}\end{pmatrix}$ and $\vec{r}=2\hat{i}+4\hat{j}+5\hat{k}+\mu\begin{pmatrix}3\hat{i}+4\hat{j}+5\hat{k}\end{pmatrix}$ is $k$, then the value of $\tan^{-1}\tan\begin{pmatrix}2\sqrt{6}k\end{pmatrix}$ is

Question 94 : If $\vec {a},\vec {b},\vec {c}$ are position vectors of the non-collinear points $A, B, C$ respectively, then the shortest distance of $A$ from $BC$ is<br/>

Question 95 : If the shortest distance between the lines $\displaystyle \frac{x-1}{\alpha} = \frac{y+1}{-1} = \frac{z}{1}, \,\,\, (\alpha \neq -1)$ and <div><br/></div><div>$x + y + z + 1 = 0 = 2x - y + z + 3$ is $\displaystyle \frac{1}{\sqrt{3}}$, then a value of $\alpha$ is :</div>

Question 96 : Find the shortest distance between the line whose equations are $\vec { r } =\hat { i } +2\hat { j } +3\hat { k } +\lambda \left( 2\hat { i } +3\hat { j } +4\hat { k } \right) $ and $\vec { r } =2\hat { i } +4 \hat {j} +5\hat { k } +\mu \left(4\hat { i } +6\hat { j } +8\hat { k } \right) $<br><br>

Question 97 : The cartesian from of equation a line passing through the point position vector $2\hat{i}-\hat{j}+2\hat{k}$ and is in the direction of $-2\hat{i}+\hat{j}+\hat{k}$, is

Question 98 : The distance of the point $P(2, 3, 4)$ from the line $1 - x = \dfrac {y}{2} = \dfrac {1}{3}(1 + z)$ is

Question 99 : Let $A(\vec a)$ and $B(\vec b)$ be points on two skew line $\vec r=\vec a+\lambda \vec p$ and $\vec r=\vec b+u\vec q$ 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 $\dfrac {1}{2}$ units. If an angle between AB and the line of shortest distance is $60^o$, then $AB=$

Question 100 : Find the shortest distance between the lines $\vec{r} = (4i - j) + \lambda (i + 2j - 3k)$ and $\vec{r} = (i - j + 2k) + \mu(2 i + 4j - 5k)$<br/>