Question 1 :
The angle between the lines whose direction cosines are $\left( \dfrac {\sqrt{3}}{4}, \dfrac {1}{4}, \dfrac {\sqrt{3}}{2} \right)$ and $\left( \dfrac {\sqrt{3}}{4}, \dfrac {1}{4}, -\dfrac {\sqrt{3}}{2} \right)$ is :
Question 2 :
The angle between the lines $\dfrac {x - 1}{1} = \dfrac {y - 1}{1} = \dfrac {z - 1}{2}$ and, $\dfrac {x - 1}{-\sqrt {3} - 1} =  \dfrac {y - 1}{\sqrt {3} - 1} = \dfrac {z - 1}{4}$ is
Question 4 :
If the lines $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z}{5t-1}$ and $\dfrac{x+1}{2s+1}=\dfrac{y}{2}=\dfrac{z}{4}$ are parallel to each other, then value of $s$, or $t$ will be<span>?<br></span>
Question 6 :
$ \xrightarrow [ A ]{ } .\left( \xrightarrow [ A ]{ } \times \xrightarrow [ B ]{ } \right) $
Question 7 :
If $|\overrightarrow a - \overrightarrow b |\overrightarrow a $ $=$ $\frac{1}{2}\left| {\overrightarrow b } \right|$ and $(\overrightarrow a - \overrightarrow b )$ is $\ \bot $ to $\overrightarrow a $ $,$ then what is the angle between $\overrightarrow a $ and $\overrightarrow b $ $?$
Question 8 : 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 9 :
If $\overset{\rightarrow }{A}\times \overset{\rightarrow }{B}=\overset{\rightarrow }{0}$ and $\overset{\rightarrow }{B}\times \overset{\rightarrow }{C}=\overset{\rightarrow }{0}$, then the angle between $\overset{\rightarrow }{A}$ and $\overset{\rightarrow }{C}$ may be :
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 :
Angle between the vectors $(\hat{i} + \hat{j})$ and $(\hat{j} - \hat{k})$ is
Question 12 :
Find the vector equation of line joining the points $ (2,1,3)$ and $(-4,3,-1)$
Question 13 :
What is the angle between $\hat {i} + \hat {j} + \hat {k}$ and $\hat {j}$?
Question 14 :
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 15 :
What is the angle between $\vec{P} \times \vec{Q}$ and $\vec{Q} \times \vec{P}$ ?
Question 17 :
Vectors $\vec {A}, \vec {B}$ and $\vec {C}$ are such that $\vec {A} \cdot \vec {B}=0$. Then the vector parallel to $\vec {A}$ is
Question 18 : 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 19 :
A straight line is equally inclined to all the three coordinate axes. Then an angle made by the line with the y-axis is
Question 20 :
Number of vectors perpendicular to vectors<span>$( 0,2,2 )$ and $( 2,2,0 )$ are $\ldots \ldots$ .</span>
Question 21 :
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 22 :
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 24 :
The value of $k$ so that the lines $\dfrac { x-1 }{ -3 } =\dfrac { y-2 }{ 2k } =\dfrac { z-3 }{ 2 }$, $\dfrac { x-1 }{ 3k } =\dfrac { y-5 }{ 1 } =\dfrac { z-6 }{ -5 }$ are perpendicular to each other, is:<br/>
Question 25 :
<b></b>Lines $  \frac{x}{2}=\frac{y}{1}=\frac{z-2}{1} $ $ \frac{x-2}{2}=\frac{y+1}{1}=\frac{3-z}{-5}  $ are $  \ldots \ldots  $ lines.
Question 26 :
Vector equation of the line $6x - 2 = 3y + 1 = 2z - 2$ is
Question 27 : 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 28 :
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 29 :
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 30 :
If the line joining the points $(-1, 2, 3), (2, -1, 4)$ is perpendicular to the line joining the points $(x, -2, 4), (1, 2, 3)$ then $x =$.
Question 31 :
If $\vec P + \vec Q = \vec P - \vec Q$ and q is the angle between $\vec P$ and $\vec Q$, then
Question 32 :
The vector equation $r=i-2j-k+t(6j-k)$ represents a straight line passing through the points:
Question 33 :
If $|A\times B|=\sqrt{3}$ A$\cdot$B then the value of $|A+B|$ is ?<br>
Question 34 :
A vector perpendicular to both vector $\overrightarrow A = \hat i + 2\hat j + \hat k $ as well as $\overrightarrow B = \hat i + \hat j - \hat k $ is
Question 35 :
If the angle between the vectors $\vec{A}$ and $\vec{B}$ is $\theta$, the value of the product ($\vec{A}\times\vec{B}) \cdot \vec{A}$ is equal to
Question 36 :
If lines <br/>$\dfrac{x+1}{1}=\dfrac{y+2}{\lambda}=\dfrac{z-1}{-1}$ and $\dfrac{x-1}{\lambda}=\dfrac{y+1}{2}=\dfrac{z+1}{1}$<br/>are perpendicular to each other than value of $\lambda$ is<br/><br/>
Question 37 :
The angle between the lines whose direction cosiones satisfy the equations $l+m+n=0$ and $l^{2}=m^{2}+n^{2}$ is
Question 38 :
If $ |\vec A \times \vec B | =\sqrt { 3 } |\vec A| | \vec B| $, then the value of $ |A + B| $ is ;
Question 39 : 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 40 : <p>The line through<br>$\widehat{i} + 3 \widehat{j} + 2 \widehat{k}$ and $\perp $ to the<br>line $\overrightarrow{r} = \left ( \widehat{i} + 2 \widehat{j}<br>- \widehat{k} \right ) + \lambda \left ( 2\widehat{i} + <br>\widehat{j} + \widehat{k} \right )$ and $\overrightarrow{r} = \left ( 2\widehat{i} + 6 \widehat{j} + \widehat{k} \right ) + \mu \left ( \widehat{i} + 2\widehat{j} + 3\widehat{k} \right )$</p>
Question 41 :
The vector equation of line passing through the point $(-1,-1,2)$ and parallel to the line $2x-2=3y+1=6z-2$
Question 42 :
Given $\overrightarrow{P}.\overrightarrow{Q}=\left | \overrightarrow{P}\times \overrightarrow{Q}  \right |$ and $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$ then $\left | \overrightarrow{R} \right | $ is:
Question 44 :
If the d.rs of two lines are given by the equations $l+m+n=0$ and $2lm-mn+2nl=0$, then the angle between the two lines is<br/>
Question 45 :
If the angle between the  vectors $ \overrightarrow A $ and $ \overrightarrow B $ is $ \theta $ , the value of the product $ (\overrightarrow B \times \overrightarrow A) . \overrightarrow A $ is equal to:
Question 46 :
If vector $\vec a = 4\hat i+5\hat j-3\hat k$ and $\vec b = 5\hat i+3\hat j+8\hat k$ then value of $\dfrac{\text{projection of vector b on a}}{\text{projection of vector a on b}}$ is :
Question 47 :
What positive value for $k$ would make the following the equations of a pair of parallel lines on the same coordinate axes?<br/><span>$y = kx $ and $ky = 9x $</span>
Question 49 :
ABC is a triangle where $A = ( 2,3,5 ) , B = ( - 1,2,2 )$ and $C (\lambda,5 , \mu )$ if the median through A is equally inclined to the positive axis then $\lambda + \mu$ is
Question 50 :
Two vectors $\vec{A}$ and $\vec{B}$ have magnitude $3$ each. $\vec{A} \times \vec{B} =-5 \hat{k}+2 \hat {i}$. Find the angle between $A$ and $B$.
Question 51 :
If the lines $x = ay+b, z = cy + d$ and $x=a'z + b',<br/>y = c'z + d'$ are perpendicular, then:<br/>
Question 52 :
Let $\theta $ be the angle between two vectors $\overrightarrow{A} $ and $\overrightarrow{B} $ then $\dfrac{\widehat{A}\times \widehat{B}}{\widehat{A}\cdot \widehat{B}}$ is equal to :<br/>
Question 53 :
If $\vec {A} \cdot \vec {B} = |\vec {A} \times \vec {B}|$, find $|\vec {A} - \vec {B}|$.
Question 54 :
If $A(3, 4, 5), B(4, 6, 3), C(-1, 2, 4)$ and $D(1, 0, 5)$ are such that the angle between the lines $\overline{DC}$ and $\overline{AB}$ is $\theta$ then $cos\,\theta =$
Question 55 :
The angle between the straight lines $\displaystyle \frac { x+1 }{ 2 } =\frac { y-2 }{ 5 } =\frac { z+3 }{ 4 } $ and $\displaystyle \frac { x-1 }{ 1 } =\frac { y+2 }{ 2 } =\frac { z-3 }{ -3 } $ is
Question 56 : <span>Find the point of intersection of the following pair of lines, assuming that the vectors $\vec a$ and $\vec b$ are not parallel.</span><div>$\displaystyle \vec r=\vec a+\mu \vec b$,<div>$\vec r=\vec b+\gamma \vec a$<br/></div></div>
Question 57 :
Equation of a line passing through the point $\hat{i}+\hat{j}-\hat{k}$ and parallel to the vector $2\hat{i}+\hat{j}{+}2\hat{k}$ is<br/>
Question 58 :
If $A\times B = B\times A$, then the angle between $A$ and $B$ is
Question 59 :
The lines $\displaystyle \frac{x+3}{-2}=\frac{y}{1}=\frac{z-4}{3}$ and $\displaystyle \frac{x}{\lambda }=\frac{y-1}{\lambda +1}=\frac{z}{\lambda +2}$ are perpendicular to each other. Then $\lambda$ is equal to
Question 60 :
If $\vec A \times \vec B=0$ and $\vec A. \vec B=-AB$, then angle between $\vec A$ and $\vec B$ is:
Question 61 :
The angle between the pair of lines $\dfrac{x-2}{2} = \dfrac{y-1}{5} = \dfrac{z+3}{-3}$ and $\dfrac{x+2}{-1} = \dfrac{y-4}{8} = \dfrac{z-5}{4}$ is
Question 62 :
Find the equation of the plane through the line $r=a+rb$, and parallel to the line $r=c+pd$, and hence obtain the shortest distance between the two lines
Question 63 :
The measure of the angle between the lines, whose direction numbers are $l,m,n$ and $m-n,n-l,l-m$ is ______
Question 64 :
A line passed through the point $A(6,2,2)$ and is parallel to the vector $(1,-2,2)$. Another line passes through the point $B(-4,0,-1)$ and is parallel to the vector $(3,-2,-2)$. The shortest distance between these two lines is<br/>
Question 65 :
If the lines $\dfrac {2x - 1}{2} = \dfrac {3 - y}{1} = \dfrac {z - 1}{3}$ and $\dfrac {x + 3}{2} = \dfrac {z + 1}{p} = \dfrac {y + 2}{5}$ are perpendicular to each other, then $p$ is equal to
Question 67 :
The angle between the lines whose direction cosines are given by $2l-m+2n=0$, $lm+mn+nl=0$ is
Question 69 :
What is the angle between $\vec{a}$ and $\vec{b}$<br/>(i) magnitude of $\vec{a}$ and $\vec{b}$ are $3$ and $4$ respectively.<br/>(ii) area of triangle made by $\vec{a}$ and $\vec{b}$ is $5$.
Question 70 :
If the line $\dfrac {x-1}{-3}=\dfrac {y-2}{2k}=\dfrac {z-3}{2}$ and $\dfrac {x-1}{3k}=\dfrac {y-5}{1}=\dfrac {z-6}{-5}$ are perpendicular to each other then $k=$
Question 71 :
The distance of the point $3\hat{i}+5\hat{k}$ from the line parallel to the vector $6\hat{i}+\hat{j}-2\hat{k}$ and passing through the point $8\hat{i}+3\hat{j}+\hat{k}$ is <br>
Question 72 :
The line $\displaystyle \frac {x - a}{4} = \frac {y - b}{5} = \frac {z - c}{0}$ is
Question 73 :
What is the angle between $\vec A$ and the resultant of $\left( \vec { A } +\vec { B } \right)$ and $\left( \vec { A } -\vec { B } \right)$
Question 74 :
The value of $\lambda $ for which the triangle $ABC$ whose vertices are $A(6,10,10),B(1,0,-5)$ and $C(6,-10,\lambda)$ is right-angled at $B$,is<br>
Question 76 :
For two vectors $\vec{A}$ and $\vec{B}$, $\vec{A}+\vec{B}=\vec{C}$ and $|\vec{A}|+|\vec{B}|=|\vec{C}|$. The angle between two vectors is:
Question 77 :
If the angle between the line $ x = \dfrac { y - 1 } { 2 } = \dfrac { z - 3 } { \lambda } $ and the plane $ x + 2 y + 3 z = 4 \text { is } \cos ^ { - 1 } \left( \sqrt { \frac { 5 } { 14 } } \right) $, then $ \lambda $ equals:-
Question 78 :
If $(2, -1, 2)$ and $(K, 3, 5)$ are the triads of direction ratios of two lines and the angle between them is $45^{\circ}$, then a value of $K$ is
Question 79 :
Distance between the two planes : $2x + 3y + 4z = 4$ and $4x + 6y + 8z = 12$ is
Question 80 :
The angle between two lines whose direction cosines satisfy the equations $n= l+m$ and $m= 2l+3n$ is<br>
Question 81 :
$\vec { A }$ and $\vec { B }$ are two vectors and $\theta$ is the angle between them, if $| \vec { A } \times \vec { B } | = \sqrt { 3 } ( \vec { A } \cdot \vec { B } )$ the value of $\theta$ is<br/>
Question 82 :
Let $\overrightarrow{p}=3a{x}^{2}\hat{i}-2\left(x-1\right)\hat{j}$ and $\overrightarrow{q}=b\left(x-1\right)\hat{i}+x\hat{j}$. If $ab<0$ then $\overrightarrow{p}$ and $\overrightarrow{q}$ are parallel for
Question 83 :
Vector $C$ is the sum of two vectors $A$ and $B$ and vector $D$ is the cross product of vectors $A$ and $B$. What is the angle between vectors $C$ and $D$?
