Question 1 : If $\bar{a}$ is unit vector, then $|\bar{a}\times \hat{i}|^2+|\bar{a}\times \hat{j}|^2+|\bar{a}\times \hat{k}|^2=$ _____________.

Question 4 : If $\vec{a} = (2, 1, -1),\vec{b} = (1,-1,0),\vec{c} = (5, -1, 1) $ ,then what is the unit vector parallel to $ \vec{a} + \vec{b} - \vec{c} $in the opposite direction ?

Question 5 : State the following statement is True or False<br/>If the starting and end points of a vector are collinear, it is known as a unit vector.

Question 6 : Find the direction angle of the vector $v=2(\cos 30^{o}\hat{i}+\sin 30^{o} \hat{j})$.

Question 7 : If $u,\ v,\ w$ are non-coplanar vector and $p,\ q$ are real numbers, then the equality $[3u\ pv\ pw]-[pv\ w\ qw]-[2w\ qv\ qu]=0$ holds for

Question 8 : For three vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$ which of the following expressions is not equal to any of remaining is

Question 9 : If the points $A$ and $B$ are $\left( 1,2,-1 \right)$ and $\left( 2,1,-1 \right)$ respectively, then $\vec { AB } $ is

Question 10 : If the position vectors of the points $A, B, C, D$ are$(0,2, 1)$, $(3,1,1),$ $(-5,3,2)$,$(2,4,1)$ respectively and if $PA+PB+PC+PD=0$ then the position vector of P is<br/>

Question 11 : Given that $\vec{ A } \times \vec{ B } =\vec{ B } \times \vec { C } =\vec { 0 } $ if $\vec{ A } \vec { B } \vec { C } $ are not null vectors, Find the value of $\vec{ A } \times \vec{ C } $

Question 13 : find the coordinate of the tip of the position vector which is equivalent to $ \vec{AB}$, where the coordinates of A and B are (-1, 3) and (-2, 1) respectively.

Question 14 : Let $ABCD$ be a parallelogram whose diagonals intersect at $P$ and $O$ be the origin, then $\vec { OA } +\vec { OB } +\vec { OC } +\vec { OD } $ equals

Question 15 : If the position vectors of the points $A(3,4),B(5, -6)$ and $C(4,-1)$ are $ \vec{a}, \vec{b}, \vec{c}$ respectively,&#160;compute $ \vec{a}+2\vec{b}-3\vec{c}. $<br/>

Question 17 : In a triangle ABC, if $ 2\vec { AC } =3\vec { CB }$, then $2\vec { OA } +3\vec { OB }$ equals ?<br/>

Question 18 : Which of the following is not a unit vector for all values of $\theta$?

Question 19 : A line passes through the points whose position vectors $ \hat { i } +\hat { j } -2\hat { k }$ and $\hat { i } -3\hat { j } +\hat { k }$. Then the position vector of a point on it at a unit distance from the first point is

Question 21 : Find a vectorin which two of the three direction angles are $\alpha=75^{o}$ and $\beta=55^{o}$.

Question 24 : Express $ \vec{AB}$ in terms of unit vectors $ \hat{i} $ and $\hat{j}$, when the points are:<br>A(4,-1), B(1,3)<br>Find $ \left | \vec{AB} \right |$ in each case.

Question 26 : At what value of the parameter n the&#160;vectors $\overrightarrow{a}=\hat{i}+2\hat{j}+4\hat{k}$ and&#160;$\overrightarrow{a}=\hat{i}+2\hat{j}+2n\hat{k}$ are equal

Question 28 : If $\left| {\widehat a - \widehat b} \right| = \sqrt 3 $ , then $\left| {\widehat a + \widehat b} \right|$ may be:-

Question 31 : Four forces act on a point object. The object will be in equilibrium, if:

Question 32 : If $\lambda (2\overline {i} - 4\overline {j} + 4\overline {k})$ is a unit vector then $\lambda =$

Question 33 : When a body is thrown up, the sign of $g$ is positive when it goes up.

Question 34 : If $|\vec{a} +\vec{b}| &gt; |\vec{a} - \vec{b}|$ then the angle between $\vec{a}$ and $\vec{b}$ is

Question 35 : A straight line is inclined to the axes of $Y$ and $Z$ at angles $45^{\circ}$ and $60^{\circ}$ respectively. The inclination of the line with the $X$-axis is<br/>

Question 36 : $\mathrm{If}$ $\vec{AD},\ \vec{BE},\ \vec{CF}$ are medians of an equilateral triangle $\mathrm{A}\mathrm{B}\mathrm{C}$, then $\vec{AD}+\vec{BE}+\vec{CF}$ equals to&#160;<br/>

Question 37 : For non zero vectors $a,b$ and $c$, if $a+b+c=0$ then which relation true:-

Question 38 : Two vectors $a$ and $b$ are said to be equal, if<br>I. $|a| = |b|$<br>II. they have same or parallel support.<br>III. the same sense.<br>Which of the following is true?

Question 39 : If $\vec{a}$ be the position vector whose tip is (5,-3), find the coordinates of a point B such that $ \vec{AB} = \vec{a},$ the coordinates of A being (4,-1).

Question 40 : Given $\vec p= (2,-4,1), \vec q = (3,-1,2), \vec r = (5,5, 4)$. Then $\vec{PQ}$ and $\vec{QR}$ are

Question 41 : Find the magnitude of two vectors $\vec a$ and $\vec b$, having the same magnitude and such that the angle between them is ${60^ \circ }$ and their scalar product is $\dfrac{1}{2}$.

Question 42 : The vectors $\hat { i } +2\hat { j } +3\hat { k }$, $2\hat { i } -\hat { j } +\hat { k }$ and $3\hat { i } +\hat { j } +4\hat { k }$ are so placed that the end point of one vector is the starting point of the next vector. Then the vectors are :

Question 45 : In Polygon Law of Vector Addition, the head of first vector is joined to the tail of last vector.

Question 46 : The Polygon Law of Vector Addition is simply an extension of ____________.

Question 47 : The vector $z&#160;= 3 - 4i$ is turned anticlockwise through an angle of&#160;$180^{\circ}$ and stretched $\dfrac{5}{2}$ times. The&#160;complex number corresponding to the newly obtained vector is ....

Question 51 : If $\vec { a }$and $\vec { b }$&#160; are not perpendicular and $\vec { c }$ and $\vec {d} $ are such that&#160;$\overrightarrow b&#160; \times \overrightarrow c&#160; = \overrightarrow b&#160; \times \overrightarrow a \,\,and\,\,\,\overrightarrow a .\overrightarrow d&#160; = 0$ then $\vec {d}$ is equal to-&#160;

Question 52 : If $\vec a$ is a non-zero vector of modulus $a$ and $m$ is a non-zero scalar, then $m \vec a$ is a unit vector if

Question 53 : If $\vec {a}$ and $\vec {b}$ are non zero vectors such that $|\vec {a} + \vec {b}| = |\vec {a} - 2\vec {b}|$, then

Question 54 : If $p$ , $q$ and $r$ are three non-coplanar vectors such that $p + q + r = \alpha s$ and $q + r + s = \beta p $. then $ p +q +r +s $ is equal to

Question 55 : The direction cosines of two lines are $(l_{1},m_{1},n_{1})$ and $(l_{2},m_{2},n_{2})$, then the value of $(l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2})^{2}+\displaystyle \sum(m_{1}n_{2}-m_{2}n_{1})^{2}$ is<br/>

Question 56 : If $\overrightarrow a =3\widehat{i}-2\widehat{j}+\widehat{k}$ and $\overrightarrow b =2\widehat{i}-4\widehat{j}-3\widehat{k}$, find $|\overrightarrow a -2\overrightarrow b|$.

Question 57 : If $\vec { x } $ is a vector in the direction of $(2,-2,1)$ of magnitude $6$ and $\vec { y } $ is a vector in the direction of $(1,1,-1)$ of magnitude $\sqrt{3}$, then $\left| \vec { x } +2\vec { y } \right| =...$

Question 59 : If $\widehat i + \widehatj, \widehatj + \widehatk, \widehati + \widehatk$ are the position vectors of the vertices of a $\Delta ABC$ taken in order, then $\angle A$ is equal to

Question 61 : Let $a,b,c$ be three non-zero vectors such that no two of these are collinear. If the vectors $a+2b$ is collinear with $c$ and $b+3c$ is collinear with $a$ ($\lambda$ being some non-zero scalar), then $a+2b+6c$ equals to

Question 62 : If the position vectors of A, B, C, D are $\vec{a}, \vec{b}. 2\vec{a}+3\vec{b}, \vec{a}-2\vec{b}$ respectively, then $\vec{AC}, \vec{DB}, \vec{BA}, \vec{DA}$ are?

Question 63 : Let $\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}, b=\hat{i}+2\hat{j}-2\hat{k}$. Then a unit vector perpendicular to both $\vec{a}-\vec{b}$ and $\vec{a}+\vec{b}$ is :

Question 64 : a and c are unit collinear vectors and $\left|b\right| = 6$, then $b-3c=\lambda a$, if $\lambda $ is

Question 65 : Let $A,B,C$ be distinct point with position vectors $\hat{i}+\hat{j}$, $\hat{i}-\hat{j}$, $p\hat{i}-q\hat{j}+r\hat{k}$ respectively. Points $A,B,C$ are collinear, then which of the following can be correct:

Question 66 : Let $OB=\hat { i } +2\hat { j } +2\hat { k }$ and $OA=4\hat { i } +2\hat { j } +2\hat { k } $. The distance of the point $B$ from the straight line passing through $A$ and parallel to the vector $2\hat { i } +3\hat { j } +6\hat { k } $ is

Question 67 : If $\alpha,\ \beta,\ \gamma$ are the angles made by a vector with the coordinate axes in the positive direction, then the range of $\sin\alpha\sin\beta+\sin\beta\sin\gamma +\sin\gamma \sin\alpha$ is<br/>

Question 68 : Assertion: $\vec{a} = 3 \vec{i} + p \vec{j} + 3 \vec{k}$ and $\vec{b} = 2 \vec{i} + 3 \vec{j } + q\vec{k}$ are parallel vectors if $p = \dfrac{9}{2}$ and $q = 2$. Reason: If $\vec{a}= a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ and $\vec{b} = b_1 \vec{i} + b_2 \vec{j} + b_3 \vec{k}$ are parallel, then $\displaystyle \dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} = \dfrac{a_3}{b_3}$

Question 69 : A system of vectors is said to be coplanar, if<br/>I. Their scalar triple product is zero.<br/>II. They are linearly dependent.<br/>Which of the following is true?

Question 70 : If the vector $p\hat i + \hat j + \hat k,\hat i + q\hat j + \hat k$ and $\hat i + \hat j + r\hat k\left( {p \ne q \ne r \ne 1} \right)$ are coplanar, then the value of $pqr - \left( {p + q + r} \right)$ is

Question 72 : Let $u, v$ and $w$ be such that $\left| u \right| =1, \left| v \right| =3$ and $\left| w \right| =2$. If the projection of $v$ along $u$ is equal to that of $w$ along $u$ and vectors $v$ and $w$ are perpendicular to each other, then $\left| u-v+w \right| $ equals

Question 73 : If the position vectors of the vertices of a triangle be 6i + 4j + 5k, 4i + 5j + 6k and 5i + 6j + 4k then the triangle is

Question 74 : If a vector make angles $\alpha,\ \beta$ and $\gamma$ with axes $x,y$ and $z$ , then $\cos 2\alpha+\cos 2\beta+\cos 2\gamma =$<br/>

Question 75 : If a, b, c are position vectors of the vertices of a $\displaystyle \Delta ABC,$ then $ \displaystyle \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=$

Question 76 : If $\vec { a } ,\vec { b } ,\vec { c } $ are mutually perpendicular unit vectors, then $\left| \vec { a } +\vec { b } +\vec { c } \right| $ is equal to

Question 77 : Let $u, v$ and $w$ be vectors such that $u + v + w = 0$. If $|u| = 3, |v| = 4$ and $|w| = 5$, then $u . v + v . w + w.u$ is equal to

Question 78 : If $a,b,c \in N$, the number of points having position vectors $a\hat i + b\hat j + c\hat k$ such that $6 \le a + b + c \le 10$ is

Question 79 : If $l,m,n$ are the direction cosines of a vector if $l=\displaystyle \dfrac{1}{2}$ ,then the maximum value of $lmn$ is<br/>

Question 80 : The vector sum of (N) coplanar forces, each of magnitude F, when each force is making an angle of<br/>$\frac{2\pi }{N}$ with that preceding one, is :<br/>

Question 81 : The resultant of $P$ and $Q$ is $R$. If $Q$ is doubled, $R$ is also doubled and if $Q$ is reversed, $R$ is again doubled. Then, $P^{2} : Q^{2} : R^{2}$ given by

Question 82 : If $a = \widehat i + 2 \widehat j + 2 \widehat k, |b|,=5$ and the angle between $a$ and $b$ is $\dfrac {\pi}{6}$, then the area of the triangle formed by these two vectors as two sides is

Question 83 : If $|a|=5.|\vec{b}|=4$, and $|c|=3$. then what will be the value of $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ given that $\vec{a}+\vec{b}+\vec{c}=0$

Question 84 : If&#160;$|\vec{a} + \vec{b}| &lt; |\vec{b} - \vec{a}|$ then angle between $\vec{a} \, $and $\, \vec{b}$ is

Question 87 : For which values of '$a$' the different vectors $ \overline { x } =\left( 2a,3a,0 \right) $ and $\overline { y } =\left( 0,0,4a \right) $ are orthogonal vectors

Question 88 : If $l, m$are the direction cosines of a line lying in the $xy$plane, then

Question 89 : If three vectors $ a, b, c $ satisfy $ a+b+c=0$ and $ |a| = 3, |b| = 5, |c| = 7 , $ then the angle between $a$ and $b$ is :

Question 91 : If $\overrightarrow{a}$ and $\overrightarrow{b}$ two collinear vectors then which of the following are incorrect<br/>

Question 92 : Let $\hat {a}$ and $\hat {b}$ two unit vector such that ${ \left( \hat { a } .\hat { b } \right) }^{ 2 }-\left| \hat { a } \times \hat { b } \right| $ is maximum then $\left| \hat { a } .\hat { b } \right|$ is equal to

Question 93 : If the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - 6\hat{k} \, and \, \hat{b} = x\hat{i} - \hat{j} + 2\hat{k}$ are parallel, then x =

Question 94 : Find the direction cosines of the vector $\vec { a } =\hat { i } +\hat { j } -2\hat { k } $.

Question 95 : Let $\vec{a},\vec{b}$ and $\vec{c}$ be non-coplanar unit vectors equally inclined to one another at an acute angle $\theta$ then $\left[\vec{a}\vec{b}\vec{c}\right]$ in terms of $\theta$ is equal to :

Question 96 : If $\bar{b} \, and \, \bar{c}$ are two non-collinear vectors such that $\bar{a} || (\bar{b} \times \bar{c})$, then $\, (\bar{a} \times \bar{b}). (\bar{a} \times \bar{c})$ is equal to&#160;

Question 97 : $x$ component of &#160;$\overline{a}$ &#160;is twice of its $y$-component. If the magnitude of the vector is $5\sqrt{2}$ and it makes an angle of $135^{\circ}$ with $z$-axis then the vector is<br/>

Question 98 : Position vector of a point $P$ is a vector whose initial point is origin.

Question 99 : If $\vec {a} + \vec {b} + \vec {c} = \vec {0}$, then which of the following is/ are correct?<br>$1.\ \vec {a}, \vec {b}, \vec {c}$ are coplanar.<br>$2.\ \vec {a} \times \vec {b} = \vec {b}\times \vec {c} = \vec {c} \times \vec {a}$<br>Select the correct answer using the code given below.

Question 100 : Let $a,b$ and $c$ be three vectors satisfying $\quad a\times b=\left( a\times c \right) $, $\left| a \right| =\left| c \right| =1,\left| b \right| =4$ and $\left| b\times c \right| =\sqrt { 15 } $. If $b-2c=\lambda a$, then $\lambda$ equals

Question 101 : Find a vector in the direction of vector $5\hat{i}-\hat{j}+2\hat{k}$ which has magnitude$8$ units.

Question 102 : If $a$ and $b$ are unit vectors and $\theta$ is the angle between them, then $a-b$ will be a unit vector if $\theta=$

Question 103 : Forces $3 \vec { OA }$, $5 \vec { OB }$ act along OA and OB. If their resultant passes through C on AB, then :<br><br>

Question 104 : <p>If the sum of two unit vectors is also a unit vector, then the angle&#160;between the two vectors is</p>

Question 105 : Let $\vec { p } $ and $\vec { q } $ be the position vectors of the points $P$ and $Q$ respectively with respect to origin $O$. The points $R$ and $S$ divide $PQ$ internally and externally respectively in the ratio $2:3$. If $\overrightarrow { OR } $ and $\overrightarrow { OS } $ are perpendicular, then which one of the following is correct?

Question 107 : If vectors $i+2j+2k$ is rotated through an angle of ${90}^{o}$ so as to cross positive direction of y-axis, then the vector in the new positive is?

Question 109 : If $a=i-j,b=i+j,c=i+3j+5k$ and $n$ is a unit vector such that $b,n=0,a,n=0$ then the value of $|c,n|$ is equal to

Question 110 : Let $\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } $ are three non zero vectors such that any two of them are non-collinear. If $\overrightarrow { a } +\overrightarrow { b } $ is collinear with $\overrightarrow { c } $ and $\overrightarrow { b } +\overrightarrow { c } $ is collinear with $\overrightarrow { a } $, then the value of $\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } $ equals

Question 111 : A unit vector a makes an angel $\Pi /4$ with the z-axis. If a+i+j is a unit vector, then a can be equal to

Question 113 : If $\bar{u}, \bar{v}, \bar{w}$ are non-coplanar vectors and p,q are real numbers, then the equality&#160;<br/>$[3 \bar{u} \, p\bar{v} \, p \bar{w}] - [p\bar{v} \, \bar{w} \, q\bar{u}] - [2 \bar{w} \, q \bar{v} \,&#160;&#160;q\bar{u}] = 0$ hold for

Question 114 : A unit vector $d$ is equally inclined at an angle $\alpha$ with the vectors $a=\cos \theta. i+ \sin \theta. j , b=-\sin \theta.i+\cos =\theta. j$ and $c=k$. Then $\alpha$ is equal to&#160;

Question 115 : If the vectors $\hat i - \hat j, \hat j + \hat k$ and $\vec a$ form a triangle, then $\vec a$ may be

Question 116 : Vectors $\vec a = \hat i + 2 \hat j + 3 \hat k, \vec b = 2 \hat i - \hat j + \hat k$ and $\vec c = 3 \hat i + \hat j + 4 \hat k$ are so placed that the end point of one vector is the starting point of the next vector, then the vectors are

Question 117 : If $\vec a,\ \vec b$ and $\vec c$ are three unit vectors, then $|\vec a-\vec b|^{2}+|\vec b-\vec c|^{2}+|\vec c-\vec a|^{2}$ does not exceed

Question 119 : The three points A, B, C with position vectors $-2a+3b+5c, a+2b+3c, 7a-c$&#160;<br/>

Question 120 : Cosine of an angle between the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ if $|\vec{a}|=2, |\vec{b}|=1$ and $\vec{a}$ ^ $\vec{b}=60^o$ is?