Question 1 :
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 2 :
If $|\overline {a}| = 3, |\overline {b}| = 4$ and $|\overline {a} - \overline {b}| = 5$ then $|a + b| =$
Question 3 :
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 4 :
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 6 :
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 10 :
Namita walks 14 metres towards west, then turns to her right and walks 14 metres and then turns to her left and walks 10 metres. Again turning to her left she walks 14 metres. What is the shortest distance (in metres) between her starting point and the present position ?
Question 11 :
Let $\overrightarrow { a } ,\overrightarrow { b }$ and $\overrightarrow { c } $ be three non-zero vectors such that no two of them are collinear and $(\overrightarrow { a } \times \overrightarrow { b } )\times \overrightarrow { c } =\frac { 1 }{ 3 } \left| \overrightarrow { b } \right| \left| \overrightarrow { c } \right| \overrightarrow { a } $. If $\theta$ is the angle between vectors $\overrightarrow { b }$ and $\overrightarrow { c }$, then a value of $\sin { \theta } $ is
Question 12 :
If $\bar a, \bar b, \bar c$ are unit vector and $\bar c=\bar a+\bar b$, then $|\bar a-\bar b|$ is
Question 13 :
Consider two vectors $\vec{F_{1}}=2\hat{i}+5\hat{k}$ and $\vec{F_{2}}=3\hat{j}+4\hat{k}$. The magnitude of the scalar product of these vectors is
Question 14 :
If $\bar{a}$ and $\bar{b}$ are non-collinear unit vectors and $\left|\bar{a}+\bar{b}\right|=\sqrt{3}$ then $\left(2\bar{a}+5\bar{b}\right)\cdot \left(3\bar{a}-\bar{b}\right)=?$
Question 15 :
Let $2\hat{i}+\hat{k}=\vec{\mathrm{a}},\ 3\hat{j}+4\hat{k}=\vec{{b}}$, $8\hat{i}-3\hat{j}$ $=\vec{\mathrm{c}}$. If $\vec{a}={x}\vec{b}+{y}\vec{{c}}$, then $(x,y) $ is equal to<br/>
Question 16 :
Let a,b be two noncoffinear vectors. If $\overline { OA } =\left( x+4y \right) \overline { a } +\left( 2x+y+1 \right) \overline { b } ,\overline { OB } =\left( y-2x+2 \right) \overline { a } +\left( 2x-3y-1 \right) \overline { b }$ and $3\overline { OA } =2\overline { OB }, $ then $\left( x,y \right) =$
Question 18 :
For $O$ being the origin and $3$ points $P,Q$ and $R$ lie on a plane. If $\displaystyle \vec{PO}+\vec{OQ}=\vec{QO}+\vec{OR}$, then $P, Q, R$ are <br/>
Question 20 :
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 21 :
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, compute $ \vec{a}+2\vec{b}-3\vec{c}. $<span><br/></span>
Question 22 :
If $\left[ \overrightarrow { a } \overrightarrow { b } \overrightarrow { c } \right] =1$ then $\frac { \overrightarrow { a } .\overrightarrow { b }\times \overrightarrow { c } }{ \overrightarrow { c }\times \overrightarrow { a } .\overrightarrow { b } } +\frac { \overrightarrow { b } .\overrightarrow { c }\times \overrightarrow { a } }{ \overrightarrow { a }\times \overrightarrow { b } .\overrightarrow { c } } +\frac { \overrightarrow { c } .\overrightarrow { a }\times \overrightarrow { b } }{ \overrightarrow { b }\times \overrightarrow { c } .\overrightarrow { a } }$ <span>is equal to</span>
Question 23 :
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 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><span>Find $ \left | \vec{AB} \right |$ in each case.</span>
Question 25 :
If $\lambda (2\overline {i} - 4\overline {j} + 4\overline {k})$ is a unit vector then $\lambda =$
Question 26 :
If $ | \overline{a} | = 1 , | \overline{b} | = 2, | \overline{a} - \overline{b} |^2 + | \overline{a} + 2 \overline{b} |^2 = 20, $ then $ ( \overline{a} , \overline{b} ) = $
Question 27 :
$\vec{A} \ and \ \vec{B}$ are two vectors, find the angle between them, if <br/>$\left | \vec{A}\times \vec{B} \right |=\sqrt{3}(\vec{A.}\vec{B})$ the value of is :-<br/>
Question 29 :
$\vec{a},\vec{b},\vec{c}$ are three non-collinear vectors such that $\vec{a}+\vec{b}$ is parallel to $\vec{c}$ and $\vec{a}+\vec{c}$ is parallel to $\vec{b}$ then:
Question 31 :
For non zero vectors $a,b$ and $c$, if $a+b+c=0$ then which relation true:-
Question 32 :
Which of the following is not a unit vector for all values of $\theta$?
Question 34 :
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 35 : <div><span>State the following statement is True or False</span><br/></div>If the starting and end points of a vector are collinear, it is known as a unit vector.
Question 38 :
$\mathrm{If}$ $\vec{AD},\ \vec{BE},\ \vec{CF}$ are medians of an equilateral triangle $\mathrm{A}\mathrm{B}\mathrm{C}$, then $\vec{AD}+\vec{BE}+\vec{CF}$ equals to <br/>
Question 39 :
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 :
If $\left| {\vec a} \right| = 2,\left| {\vec b} \right| = 3$ and $\left| {2\vec a - \vec b} \right| = 5,$ then $\left| {2\vec a + \vec b} \right|$ equals:
Question 42 :
If $2\overline a - 4\widehat i - 2\widehat j + \widehat k = 0$ then find $\overline a $.
Question 43 :
<span>If $\left| {\widehat a - \widehat b} \right| = \sqrt 3 $ , then $\left| {\widehat a + \widehat b} \right|$ may be:-</span>
Question 44 :
Let $a=\hat{i}+2\hat j+3\hat k$ and $b=3\hat i+\hat j$. Find the unit vector in the direction of the $a+b$.
Question 45 :
If a line has direction ratios $2,-1,-2$, determine its direction cosines.
Question 47 :
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 48 :
If $\left| \overrightarrow { a } \right| =7,\ \left| \overrightarrow { b } \right| =11,\ \left| \overrightarrow { a } +\overrightarrow { b } \right| = 10\sqrt 3$, then $\left| \overrightarrow { a } -\overrightarrow { b } \right| =$
Question 49 :
In a triangle ABC, if $ 2\vec { AC } =3\vec { CB }$, then $2\vec { OA } +3\vec { OB }$ equals ?<br/>
Question 51 :
Assertion: $\displaystyle\left ( A \right ): \overline{GA} +\overline{GB}+\overline{GC}=\bar{0}$ where $G$ is the centroid of triangle ABC.
Reason: $\displaystyle\left ( R \right ): \overline{AB}=$ P.V of $B-$ P.V of $A.$
Question 52 :
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 53 :
If the points $A(2, 1, 1), B(0, -1, 4)$ and $C(k, 3, -2)$ are collinear, then $k =$ _____
Question 54 :
Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+3\vec{c}$ is collinear with $\vec{a}(\lambda$ being some non-zero scalar), then $\vec{a}+2\vec{ b}+6\vec{c}$ equals
Question 55 :
If $\vec{r}=3\hat{i}+2\hat{j}-5\hat{k},\vec{a}=2\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-2\hat{k},\ \vec{c}=-2\hat{i}+\hat{j}-3\hat{k}$ such that $\vec{r}=\lambda\vec{a}+\mu\vec{b}+v\vec{c}$, then $\mu,\ \displaystyle \frac{\lambda}{2}$ , $v$ are in<br/>
Question 56 :
If point $O$ is the centre of a circle circumscribed about a triangle $ABC$. Then $\overline{OA}\sin 2A+\overline{OB}\sin2B+\overline{OC}\sin 2C=$<br/>
Question 57 :
If $\vec {a}$ and $\vec {b}$ are unit vectors, then angle between $\vec {a}$ and $\vec {b}$ for $\sqrt {3} \vec {a} - \vec {b}$ to be unit vector is
Question 58 :
If $\vec { a }$ and $\vec { b }$ are two unit vector and $\theta$ is the angle between them, then $\left( \vec { a } +\vec { b } \right)$ is a unit vector if $\theta =$
Question 59 :
lf $\overline{a},\overline{b},\ \overline{c}$ are three non-zero and non-null vectors and $\overline{r}$ is any vector in space, then $[\overline{b}\overline{c}\overline{r}]\overline{a}+[\overline{c}\overline{a}\overline{r}]\overline{b}+[\overline{a}\overline{b}\overline{r}]\overline{c}$ is equal to<br/>
Question 60 :
Given that P = 12, Q = 5 and R = 13 also $\vec P + \vec Q = \vec R$, then the angle between $\vec P$ and $\vec Q$ will be :
Question 61 :
If $\overrightarrow { a } $ is vector of magnitude $x$ , $m$ is non-zero scalar and $m\overrightarrow { a } $ is a unit vector then x in terms of m is:
Question 62 :
Let $\vec{a}, \vec{b}$ and $\vec{c}$ br non-coplanar unit vectors equally inclined to one another at an acute angle $\theta$. Then $[\vec{a}\ \vec{b}\ \vec{c}]$ in terms of $\theta$ is equal to
Question 63 :
If $\vec {a},\vec {b}$ are two vectors then which of the following statements is/are correct ?<br/>
Question 64 :
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 65 :
If $\hat{n}={a}\hat{i}+{b}\hat{j}$ is perpendicular to the vector $(\hat{i}+\hat{j})$, then the value of ${a}$ and ${b}$ may be:
Question 66 :
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 67 :
The position vectors of A, B are a, 6 respectively. The position vector of C is $\dfrac {5\bar{a}}{3} -\bar{b}$. Then 3
Question 68 :
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 69 :
Find the direction cosines of the vector joining the points $P(1,2,-3)$ and $Q(-1,-2,1)$, which is directed from $P$ to $Q$.
Question 70 :
The value of $x$ if $x(\hat { i } +\hat { j } +\hat { k } )$ is a unit vector is
Question 71 :
A point $O$ is the centre of a circle circumscribed about a triangle $ABC$, then $\vec{OA}\sin 2A + \vec{OB}\sin 2B + \vec{OC} \sin 2C $ is equal to
Question 72 : <div><span>Given that the vectors $\bar a,\bar b,\bar c$ form a base, find the sum of co-ordinates of the vector: </span><span>$3\bar u-\bar v+\bar w$ </span></div>if $\bar u=\bar a+\bar c, \bar v=\bar b+\bar c,\bar w=\bar a-\bar b$;
Question 73 :
<span>If two or more vectors are parallel to the same line, irrespective of their magnitudes and directions, then they are</span><br/>
Question 75 :
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 76 :
If $\vec{e}=l\hat{i}+m \hat{j}+n\hat{k}$ is a unit vector, then the maximum value of $lm+mn+nl$ is<br/>
Question 77 :
If $\vec a = \widehat i + 2 \hat j + 3\hat k, \vec b = 2 \hat i + 3 \hat j + \hat k, \vec c = 3 \hat i + \hat j + 2 \hat k$ are vectors satisfying<br/>$\alpha\vec a + \beta \vec b + \gamma \vec c = - 3 (\hat i - \hat k)$, then the ordered triplet $(\alpha, \beta, \gamma)$ is
Question 78 :
If $|\vec{a}|=5, |\vec{a}-\vec{b}|=8$ and $|\vec{a}+\vec{b}|=10$, then $|\vec{b}|$ is equal to :
Question 79 :
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 80 :
If $\overrightarrow{a}$ and $\overrightarrow{b}$ two collinear vectors then which of the following are incorrect<br/>
Question 81 :
Let $\displaystyle \vec{\alpha} =\left ( x+4y \right )\vec{a}+\left ( 2x+y+1 \right )\vec{b}$ and $\vec{\beta} =\left ( y-2x+2 \right )\vec{a}+\left ( 2x-3y-1 \right )\vec{b}$ where $\vec{a}$ and $\vec{b}$ are non-zero, non-collinear. If $\displaystyle 3\vec{\alpha}=2\vec{\beta}$ then
Question 82 :
The point with position vectors $(2, 6), (1, 2)$ and $(p, 10)$ are collinear if the value of $p$ is
Question 83 :
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 87 :
If vectors $\displaystyle \left ( x-2 \right ) \vec{a}+\vec{b}$ and $\displaystyle\left ( 2x+1 \right ) \vec{a}-\vec{b}$ are parallel then $x$
Question 88 :
$x$ component of $\overline{a}$ 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 89 :
If a vector is multiplied by a real number, then which of the following statements is incorrect?
Question 90 :
The vector that must be added to the vector $\hat{i}-3\hat{j}+2\hat{k}$ and $3\hat{i}+6\hat{j}-7\hat{k}$ so that the resultant vector is a unit vector along the y-axis is:
Question 91 :
If the position vector $\overrightarrow{a}$ of the point $(5, n)$ is such that $|\overrightarrow{a}|=13$, then the value/values of n be
Question 92 :
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 93 :
In a trapezium the vector $\overline{BC} = \alpha \overline{AD}$. We will then find that $\bar{p}= \overline{AC}+\overline{BD}$ is collinear with $\overline{AD}$. if $\bar{p}= \mu \overline{AD}$ then <br>
Question 94 :
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 95 :
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 96 :
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 97 :
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 98 :
Let us define, the length of a vector $a\overline{i}+b\overline{j}+c\overline{k}$ as $|{a}|+|{b}|+|{c}|$. This definition coincides with the usual definition of the length of a vector $a\overline{i}+b\overline{j}+c\overline{k}$ if<br/>
Question 99 :
If $O$ and $O'$ are circumcenter and orthocenter of a triangle $ABC$ then $\left( OA+OB+OC \right) $ equals
Question 100 :
Value of x and y so that the vectors $2 \overrightarrow i + 3 \overrightarrow j$ and $x \widehat i + y \widehat j$ are equal :<br/>
