Question 1 :
The direction cosines of the vectors $2\vec {i} + \vec {j} - 2\vec {k}$ is equal to
Question 2 :
The direction angles of the line $x = 4z + 3, y = 2 - 3z$ are $\alpha, \beta$ and $\gamma$, then $\cos \alpha + \cos \beta + \cos \gamma =$ ________.
Question 3 :
If $\bar {a}, \bar {b}$ and $\bar {c}$ are non-zero non collinear vectors and $\theta(\neq 0 , \pi)$ is the angle between $\bar {b}$ and $\bar {c}$ if $(\bar {a}\times \bar {b}) \times \bar {c}=\dfrac {1}{2} |\bar {b}|\bar {c}|\bar {a}$. then $\sin \theta =$
Question 4 :
$l = m =n = 1$ represents the direction cosines of <br/><br/>
Question 5 :
Direction cosines $l, m, n$ of two lines are connected by the equation $l-5m+3n=0$ and $7l^{2}+5m^{2}-3n^{2}=0$. The direction cosines of one of the lines are
Question 6 :
Direction cosines of the line $\cfrac { x+2 }{ 2 } =\cfrac { 2y-5 }{ 3 } ,z=-1$ are ____
Question 7 :
The points with position vectors $60\hat{i}+3\hat{j}$, $40\hat{i}-8\hat{j}$, $a\hat{i}-52\hat{j}$ are collinear if
Question 8 : <table class="table table-bordered"><tbody><tr><td> List I</td><td>List II </td></tr><tr><td><span>1) d.c's of $x -$ axis</span></td><td><span>a) $(1,1,1)$</span> </td></tr><tr><td><span>2) d.c's of $y -$ axis</span></td><td><span>b)$\left(\displaystyle \frac{]}{\sqrt{3}}\frac{]}{\sqrt{3}},\frac{]}{\sqrt{3}}\right)$</span></td></tr><tr><td><span>3) d.c's of $z -$ axis</span></td><td><span>c) $(1,0,0)$</span><br/></td></tr><tr><td><span>4) d.c's of a line makes </span><span>equal angles with axes</span></td><td><span>d) $(0,1,0)$</span></td></tr><tr><td> </td><td><span>e) $(0,0,1)$</span></td></tr></tbody></table>The correct order for 1, 2, 3, 4 is
Question 10 : <span>From the point $P(3, -1, 11)$, a perpendicular is drawn on the line $L$ given by the equation $\dfrac {x}{2} = \dfrac {y - 2}{3} = \dfrac {z - 3}{4}$. Let $Q$ be the foot of the perpendicular.</span><div>What are the direction ratios of the line segment $PQ$?</div>
Question 11 :
The direction cosines of a line which is equally inclined to axes, is given by
Question 12 :
If $P(x, y, z)$ moves such that $x=0, z=0$, then the locus of $P$ is the line whose d.cs are<br/>
Question 13 :
The points with position vectors $ 60i + 3j, 40i -8j$ and $ ai -52j $ are collinear if<br>
Question 14 :
The points $i + j + k, \, i + 2j, \, 2i+2j+k,\, 2i+3j+2k$ are
Question 15 :
A line makes an angle $\alpha,\beta,\gamma$ with the $X,Y,Z$ axes. Then $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=$<br/>
Question 16 :
Direction cosines of ray from $P(1, -2, 4)$ to $Q(-1, 1, -2)$ are
Question 17 :
The straight line $\displaystyle \frac{x - 3}{3} = \frac{y - 2}{1} = \frac{z - 1}{0}$ is
Question 18 :
Can $\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}, \dfrac{-2}{\sqrt{3}}$ be the direction cosines of any directed line?
Question 20 :
A line passes through the points (6, -7, -1) and (2, -3, 1). What are the direction ratios of the line ? <span><br></span>
Question 21 :
A vector is equally inclined to the $x$-axis, $y$-axis and $z$-axis respectively, its direction cosines are
Question 22 :
If a line makes the angles $ \alpha , \beta$ and $\gamma$ with the axes, then what is the value of $1+\cos 2\alpha +\cos 2\beta+\cos 2\gamma$<span> equal to ?</span>
Question 23 :
What are the DR's of vector parallel to $\left( 2,-1,1 \right) $ and $\left( 3,4,-1 \right) $?
Question 24 :
<br/>lf a line makes $\dfrac{\pi }{3},\dfrac{\pi }{4}$ with the $x$-axis, ${y}$-axis respectively, then the angle made by that line with the $z$- axis is<br/>
Question 25 :
If the lines $x=1+a,y=-3-\lambda a,z=1+\lambda a$ and $x=\cfrac { b }{ 2 } ,y=1+b,z=2-b$ are coplanar, then $\lambda$ is equal to
Question 26 :
The projection of the join of the two points $(1,4,5), (6,7,2)$ on the line whose d.s's are $(4,5,6)$ is
Question 27 :
If the $d.c's$ of two lines are connected by the equations $l + m + n = 0, l^2 + m^2 - n^2 = 0$, then angle between the lines is
Question 28 :
The direction cosines of a line segment AB are $ - \dfrac{2}{{\sqrt {17} }},\dfrac{3}{{\sqrt {17} }}, - \dfrac{2}{{\sqrt {17} }}$. If $AB=\sqrt {17} $ and the coordinates of A are $(3,-6,10)$, then the coordinates of B are 
Question 29 :
The direction`cosines of a line equally inclined to three mutually perpendicular lines having D.C.'s as ${\ell _1}{m_1}{n_1}:{\ell _2}{m_2}{n_2}:{\ell _3}{m_3}{n_3}\,\,$ are 
Question 30 :
In a line $OP$ through the origin $O$ makes angles of ${90^ \circ },\,{60^ \circ }\,and\,{60^ \circ }$ with $x, y$ and $z$ axis respectively then the direction cosines of $OP$ are  
Question 31 :
If $l,m,n$ are d.c's of vector $\overline {OP}$ then maximum value of $lmn$ is
Question 32 :
The equation to the altitude of the altitude triangle formed by $\left( {1,1,1} \right).\left( {1,2,3} \right),\left( {2, - 1,1} \right)$ through $\left( {1,1,1} \right)$ is
Question 33 :
A line making angles $45^o$ and $60^o$ with the positive direction of $x-$ axis and $y-$ axis respectively. Then the angle made by the line with positive direction of $z-$ axis is
Question 34 :
If the direction cosine of a directed line be $a, 3a, 7a$ then $a =$
Question 35 :
A lines makes angles $\dfrac{\alpha }{2},\dfrac{\beta }{2},\dfrac{\gamma }{2}$ with positive direction of coordinate axes, then $\cos \alpha  + \cos \beta  + \cos \gamma $ is equal to
Question 36 :
The direction cosines of the line which is perpendicular to the lines with direction cosines proportional to $(1, -2, -2)$ & $(0, 2, 1)$ are
Question 37 :
The direction cosines of two lines are related by $l+m+n=0$ and $al^2+bm^2+cn^2=0$. The lines are parallel if
Question 38 :
A line passes through the points $(6,-7,-1)$ and $(2,-3, 1)$. If the angle a which the line makes with the positive direction of x-axis is acute, the direction cosines of the line are.
Question 39 :
If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is
Question 40 : <div><span>$P\left(1,1,1\right)$ and $Q\left(\lambda,\lambda,\lambda\right)$ are two points in the space such that $PQ=\sqrt{27},$ the value of $\lambda$ can be <br/></span></div>
Question 41 :
The projection of the join of the points $(3,4,2),(5,1,8)$ on the line whose d.c's are $\left( {\frac{2}{7},\frac{3}{7},\frac{6}{7}} \right)$ is
Question 42 :
The direction cosines to two lines at right angles are (1,2,3) and (-2,$\frac{1}{2}$,$\frac{1}{3}$), then the direction cosine perpendicular to both given lines are:
Question 43 :
The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^{2}+m^{2}+n^{2}$ is
Question 44 :
$A=(-1, 2, -3), B=(5, 0, -6), C=(0, 4, -1)$ are the vertices of a triangle. The d.c's of the internal bisector of $\angle$BAC are?
Question 45 :
If the line $\vec{OR}$ makes angles $\theta_1, \theta_2, \theta_3$ with the planes $XOY, YOZ, ZOX$ respectively, then $\cos^2\theta_1+\cos^2\theta_2+\cos^2\theta_3$ is equal to
Question 46 :
A vector $\vec{V}$ is inclined at equal angles to axes OX, OY and OZ. If the magnitude of $\vec{V}$ is $6$ units, then $\vec{V}$ is?
Question 47 :
The direction cosines of the line segment joining points $(-3, 1, 2)$ and $(1, 4, -10)$ is.
Question 48 :
If $\left(\dfrac {1}{2},\dfrac {1}{3},n\right)$ are the direction cosines of a line then the value of $n$ is
Question 50 :
$\dfrac { x - 2 } { 1 } = \dfrac { y - 3 } { 1 } = \dfrac { z - 4 } { - 1 }$ & $\dfrac { x - 1 } { k } = \dfrac { y - 4 } { 2 } = \dfrac { z - 5 } { 2 }$ are coplanar then k=?<br/>
Question 51 :
If the directions cosines of a line are $ k, k, k, $ then
Question 52 :
In each of the following find the value of $k$, for which the points are collinear.<br/>(i) $(7,-2)$, $(5,1)$, $(3,k)$<br/>(ii) $(8,1)$, $(k,-4)$, $(2,-5)$<br/>
Question 54 :
The points, whose position vectors are $60i + 3j, 40i - 8j$ and $ai - 52j$ collinear, if
Question 55 :
If points $\hat i + \hat j, \hat i - \hat j$ and $p \hat i + q \hat j + r \hat k$ are collinear, then
Question 56 :
The points with position vectors $60 \widehat{i} + 3 \widehat{j}$, $40 \widehat{i} - 8 \widehat{j}$, $a\widehat{i} -52\widehat{j}$ are collinear if
Question 57 :
The three points whose position vectors are $\overline{i}+2\overline{j}+3\overline{k,}$ $3\overline{i}+4\overline{j}+7\overline{k,}$ and  $-3\overline{i}-2\overline{j}-5\overline{k}$<br/>
Question 58 :
The position vectors of three points are $2\vec{a}-\vec{b}+3\vec{c}$, $\vec{a}-2\vec{b}+\lambda \vec{c}$ and $\mu \vec{a}-5\vec{b}$ where $\vec{a}, \vec{b}, \vec{c}$ are non coplanar vectors, then the points are collinear when
Question 59 :
If the points whose position vectors are $2\overline{i}+\overline{j}+\overline{k},\ 6\overline{i}-\overline{j}+2\overline{k}$ and $14\overline{i}-5\overline{j}+p\overline{k}$ are collinear then the value of $\mathrm{p}$ is<br/>
Question 60 :
If the points with position vectors $60\hat{i}+3\hat{j}, 40\hat{i}-8\hat{j}$ and $a\hat{i}-52j$ are collinear, then $a=?$
Question 61 :
Assertion ($A$): The points with position vectors $\overline{a},\overline{b},\overline{c}$ are collinear if $2\overline{a}-7\overline{b}+5\overline{c}=0$.<br/>Reason ($R$): The points with position vectors $\overline{a},\overline{b},\overline{c}$ are collinear if $l\overline{a}+m\overline{b}+n\overline{c}=\overline{0}$.<br/>
Question 62 :
If $\vec { a } ,\vec { b } ,\vec { c } $ are three non-zero vectors, no two of which are collinear and the vector $\vec { a } +\vec { b } $ is collinear with $\vec { c }, \vec { b } +\vec { c } $ is collinear with $\vec {a},$ then $\vec { a } +\vec { b } +\vec { c }$ is equal to -
Question 63 :
If the points $\bar a + \bar b,\bar a - \bar b,\bar a + k\bar b$ are collinear, then  
Question 64 :
The position vectors of three points are $2\overrightarrow { a } -\overrightarrow { b } +3\overrightarrow { c } ,\overrightarrow { a } -2\overrightarrow { b } +\lambda \overrightarrow { c } $ and $\mu \overrightarrow { a } -5\overrightarrow { b } $, where $\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } $ are non-coplanar vectors. The points are coliinear when
Question 65 :
Find the value of $p$ for which the points $(-5,1)$, $(1,p)$ and $(4, -2)$ are collinear.<br/>
Question 66 :
Are the points (1, 1), (2, 3) and (8, 11) collinear ?
Question 67 :
The points $(k -1, \ k +2), (k, \ k +1), (k +1, \ k)$ are collinear for <span><br/></span>
Question 68 :
The vectors $2\hat{i}+3\hat{j};5\hat{i}+6\hat{j};8\hat{i}+\lambda\hat{j}$ have their initial points at $(1,1 )$. The value of  $\lambda$ so that the vectors terminate on one straight line is<br/>
Question 69 :
Three district points A, B and C with p.v.s. and $\displaystyle \vec { a } ,\vec { b } $ and $\displaystyle \vec { c } $ respectively are collinear if there exist non-zero scalars x, y, z such that
Question 70 :
If $\overline{a}$ and $\overline{b}$ are two non-collinear vectors, then the points $l_{1}\overline{a}+m_{1}\overline{b}$, $  l_{2}\overline{a}+m_{2}\overline{b}$ and $l_{3}\overline{a}+m_{3}\overline{b}$ are collinear if<br/>
Question 71 :
$\bar a,\bar b,\bar c$ are three non-zero vectors such that any two of them are non-collinear. If $\bar a+\bar b$ is collinear with $\bar c$ and $\bar b+\bar c$ is collinear with $\bar a$, then what is their sum?
Question 72 :
The values of $a$ for which point $(8, -7, a), (5, 2, 4)$ and $(6, -1, 2)$ are collinear.
Question 73 :
If the three points $A(1,6), B(3,-4)$ and $C(x,y)$ are collinear, then the equation satisfying by $x$ and $y$ is
Question 74 :
The line passes through the points $\left ( 5,1,a \right )$ & $\left ( 3,b,1 \right )$ crosses the $yz$ plane at the point $\displaystyle \left ( 0,\frac{17}{2},-\frac{13}{2} \right )$ ,then
Question 75 :
The line passing through the points $(5,1,a)$ and $(3,b,1)$ crosses the $yz$-plane at the point $\left (0,\dfrac {17}{2}, \dfrac {-13}{2}\right)$, t<span>hen </span>
Question 76 :
If the points $A(1,2,-1)$, $B(2,6,2)$ and $\displaystyle C\left ( \lambda,-2,-4 \right )$ are collinear, then $\displaystyle \lambda $ is<br/>
Question 77 :
Given $A(1,-1,0)$; $B(3,1,2)$; $C(2,-2,4)$ and $D(-1,1,-1)$ which of the following points neither lie on $AB$ nor on $CD$?<br>
Question 78 :
If a line makes angles $\alpha, \beta, \gamma$ with positive axes, then the range of $\sin{\alpha}\sin{\beta}+\sin{\beta} \sin{\gamma} +\sin{\gamma} \sin {\alpha}$ is
Question 79 :
If the points whose position vectors are $2i+j+k, 6i-j+2k$ and $14i-5j+pk$ are collinear, then the value of p is?
Question 80 :
A line in the 3-dimensional space makes an angle $\theta \left(0 < \theta \leq \dfrac {\pi}{2}\right)$ with both the x and y axes, then the set of all values of $\theta$ is the interval :
Question 82 :
If $AB=21,\ B\equiv (-2,1,-8)$ and the direction cosines of $AB$ are $\dfrac{6}{7},\dfrac{2}{7},\dfrac{3}{7}$ then the coordinates of $A$ are
Question 83 :
If the points $a(cos \alpha + i sin \alpha)$ , $b(cos \beta + i sin \beta)$ and $c(cos \gamma + isin \gamma)$ are collinear then the value of $|z|$ is: <br>( where ${z = bc \ sin(\beta-\gamma) + ca \ sin(\gamma-\alpha) + ab \ sin(\alpha - \beta) + 3i -4k}$ )<br>
Question 84 :
If a ray makes angles $\alpha, \beta, \gamma$ and $\delta$ with the four diagonals of a cube and<br>$\mathrm{A}:\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+\cos^{2}\delta$<br>$\mathrm{B}:\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma+\sin^{2}\delta$<br>$\mathrm{C}:\cos 2\alpha+\cos 2\beta+\cos 2\gamma+\cos 2\delta$<br>Arrange $A,B,C$ in descending order<br>
Question 85 : <span>Find the unit vectors perpendicular to the following pair of vectors:</span><div>$2i+j+k$, <span>$i-2j+k$</span></div>
Question 86 :
If $\alpha,\beta,\gamma\in[0,2\pi]$, then the sum of all possible values of $\alpha, \beta,\gamma$ if $\sin \alpha=-\dfrac{1}{\sqrt{2}}$, $\cos \beta=-\dfrac{1}{2}$, $\tan \gamma=-\sqrt{3}$, is
Question 87 :
If $\displaystyle l_{1},m_{1},n_{1}$ and $\displaystyle l_{2},m_{2},n_{2}$ are D.C.'s of the two lines inclined to each other at an angle $\displaystyle \theta $, then the D. C.'s of the internal and external bisectors of the angle between these lines are
Question 88 :
The direction ratios of the normal to the plane through $(1,0,0)$ and $(0,1,0)$ which makes an angle of $\dfrac{\pi }{4}$ with the plane $x + y = 3$ are-
Question 89 :
A plane mirror is placed at the origin so that the direction ratios of its normal are $(1,-1,1)$. A ray of light, coming along the positive direction of the x-axis, strikes the mirror. The direction cosines of the reflected ray are
Question 90 :
lf $\alpha,\ \beta,\ \gamma$ are the angles made by a line with the coordinate axes in the positive direction, then the <span>range of $\sin\alpha\sin\beta+\sin\beta\sin\gamma+\sin\gamma\sin\alpha$ is<br/></span>
Question 91 : Assertion ($A$): <div>Three points with position vectors $\vec{a},\vec{b},\ \vec{c}$ are collinear if $\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=\vec{0}$<span><br/><br/></span></div><div><span>Reason ($R$):</span></div><div><span>Three points ${A}, {B},\ {C}$ are collinear if $\vec{AB}={t}\ \vec{BC}$, where ${t}$ is a scalar quantity.<br/></span></div>
Question 92 :
If the position vectors of the points $A$, $B$, and $C$ be $i + j $ , $i - j$ and $ai + bj+ ck$ respective;y , then the points $A$, $B$ and $C$ are collinear if:
Question 93 :
A line makes equal angles with the coordinate axis. The direction cosines of this line are
Question 94 :
The direction cosine of a line equally inclined to the axes are
Question 95 :
If $A$ , $B$ and $C$ are three collinear points, where $A= i + 8 j - 5k $, $ B  = 6i-2j$ and $C= 9i + 4j - 3 k$, then $B$ divides $AC$ in the ratio of :
Question 96 :
The projections of a line segment on $x, y, z$ axes are $12, 4, 3$. The length and the direction cosines of the line segments are
Question 97 :
The projection of a directed line segment on the co-ordinate axes are $12, 4, 3$, the DC's of the line are<br/>
Question 98 :
lf a line makes angles $60^{o}, 45^{o}, 45^{o}$ and $\theta$ with the four diagonals of a cube, then $\sin^{2}\theta =$<br/>
Question 99 :
The direction ratios of the diagonal of a cube which joins the origin to the opposite corner are (when the three concurrent edges of the cube are coordinate axes)
Question 100 :
If points $P\left( 4,5,x \right) ,Q\left( 3,y,4 \right) $ and $ R\left( 5,8,0 \right) $ are colinear, then the value of $x+y$ is
