Question 2 :
If a line has direction ratios $2,-1,-2$, determine its direction cosines.
Question 3 : <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 4 :
The vector $T + 2 \overline { y } + 2 k$ restated through an angle $\theta$ and doubled in magnitude then it becomes $2 T + ( 2 x + 2 ) \} + ( 6 x - 2 ) k$ values of $x$ are
Question 5 :
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 6 :
Which of the following is not a unit vector for all values of $\theta$?
Question 7 :
If $\cos \alpha, \cos \beta$ and $\cos \gamma$ are direction cosines of a vector, then they satisfy which of the following ? Prove it.
Question 8 :
$\vec{A} \ and \ \vec{B}$ 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 9 :
$\mathrm{If}$ $\vec{AD},\ \vec{BE},\ \vec{CF}$ are medians of an equilateral triangle $\mathrm{A}\mathrm{B}\mathrm{C}$, then $\vec{AD}+\vec{BE}+\vec{CF}$ equals to <br/>
Question 10 :
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 13 :
If $|\overrightarrow{a}| = 5, |\overrightarrow{a} - \overrightarrow{b}|=8$ and $|\overrightarrow{a} + \overrightarrow{b}| = 10$, then $|\overrightarrow{b}|$ is equal to:
Question 14 :
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 15 :
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 16 :
The sum of two forces is $18$ N and resultant whose direction is at right angles to the smaller force is $12$N. The magnitude of the two forces are
Question 17 :
<span>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 } $</span>
Question 18 :
If $\vec{a}, \vec{b}, \vec{c}$ are three non coplanar vectors, then $(\vec{a}+\vec{b}+\vec{c})[(\vec{a}+\vec{b}) \times (\vec{a}+\vec{c})] $ is :
Question 19 :
<span>If $\left| {\widehat a - \widehat b} \right| = \sqrt 3 $ , then $\left| {\widehat a + \widehat b} \right|$ may be:-</span>
Question 21 :
Given $\vec p= (2,-4,1), \vec q = (3,-1,2), \vec r = (5,5, 4)$. Then $\vec{PQ}$ and $\vec{QR}$ are
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 24 :
Direction angle of a vector is $30^{o}$, then find the vector.
Question 25 :
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 27 :
If the points $A$ and $B$ are $\left( 1,2,-1 \right)$ and $ \left( 2,1,-1 \right)$ respectively, then $ \vec { AB } $ is
Question 29 :
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 30 :
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 32 :
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 35 :
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 36 :
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 37 :
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 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 40 :
If $2\overline a - 4\widehat i - 2\widehat j + \widehat k = 0$ then find $\overline a $.
Question 41 :
In a triangle ABC, if $ 2\vec { AC } =3\vec { CB }$, then $2\vec { OA } +3\vec { OB }$ equals ?<br/>
Question 42 :
$\vec{a},\vec{b},\vec{c}$ 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 43 :
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 44 :
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 45 :
If $\bar { OP } =2\hat { i } +3\hat { j } -\hat { k } $ and $\bar { OQ } =3\hat { i } -4\hat { j } -2\hat { k } $ then the modulus $\bar { PQ } $ is
Question 46 :
If $|\overline {a}| = 3, |\overline {b}| = 4$ and $|\overline {a} - \overline {b}| = 5$ then $|a + b| =$
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 :
For non zero vectors $a,b$ and $c$, if $a+b+c=0$ then which relation true:-
Question 50 :
If $\lambda (2\overline {i} - 4\overline {j} + 4\overline {k})$ is a unit vector then $\lambda =$
Question 51 :
$P$ is a point on the line through the point $A$ whose position vector is $\overrightarrow{a}$ and the line is parallel to the vector $\overrightarrow{b}$. If $PA=6$, the position vector of $P$ is
Question 52 :
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 53 :
If $A,B,C,D$ be any four points and $E$ and $F$ be the mid-points of $AC$ and $BD$, respectively, then $\vec{AB}+\vec{CB}+\vec{CD}+\vec{AD}$ is equal to
Question 54 :
If $\mathrm{O}$ is the circumcentre and $\mathrm{O}^{'}$ is the orthocentre of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ and if $\mathrm{A}\mathrm{P}$ is the circumdiameter then<br/>$\vec{\mathrm{A}\mathrm{O}}+\vec{\mathrm{O}^{'}\mathrm{B}}+\vec{\mathrm{O}^{'}\mathrm{C}}=$<br/>
Question 55 :
The magnitude of the scalar $p$ for which the vector $p\left( -3\hat { i } -2\hat { j } +13\hat { k } \right) $ is of unit length is:
Question 57 :
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
Question 58 :
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 59 :
If $\overline {OA}=i+j+k, \overline {AB}=3i-2j+k,\overline {BC}=i+2j-2k$ and $\overline {CD}=2i+j+3k $ then find the vector $\overline{OD}$.
Question 60 :
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 61 :
Given that the vectors $\bar{a}$ and $\bar{b}$ are non-collinear, the values of $x$ and $y$ for which the equality $2\bar{u}-\bar{v}= \bar{w}$ holds where $\bar{u}= x\bar{a}+2y\bar{b}, \bar{v}= -2y\bar{a}+3x\bar{b}$ and $ \bar{w}=4\bar{a}-2\bar{b}$ are<br>
Question 63 :
$P$ is any point on the circumcircle of $\triangle ABC$ other than the vertices. $H$ is the orthocenter of $\triangle ABC,M$ is the mid-point of $PH$ and $D$ is the mid-point of $BC$. Then
Question 64 :
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 65 :
The value of $x$ if $x(\hat { i } +\hat { j } +\hat { k } )$ is a unit vector is
Question 66 :
If $|\vec {a}| = 3, |\vec {b}| = 4$ and $|\vec {a} - \vec {b}| = 7$ then $|\vec {a} + \vec {b}| =$
Question 67 :
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 68 : <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 69 :
Let $\displaystyle \overrightarrow{OA}=a$ and $\displaystyle \overrightarrow{OB}=b$ and $\displaystyle \overrightarrow{OC}=a+b$. What is the type of the quadrilateral $OACB$?<br/>
Question 70 :
The vector having initial and terminal points as $(2, 5, 0)$ and $(-3, 7, 4)$, respectively is
Question 71 :
If $G$ is the centroid of a $\Delta ABC$, then $\vec{GA} + \vec{GB} + \vec{GC}$ is equal to
Question 72 :
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 73 :
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 74 :
If $l, m$ are the direction cosines of a line lying in the $xy$ plane, then
Question 75 :
Find the direction cosines of the vector $\vec { a } =\hat { i } +\hat { j } -2\hat { k } $.
Question 76 :
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 77 :
A unit vector along the direction $\hat i + \hat j + \hat k$ has a magnitude:
Question 78 :
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 79 :
If the points $A(2, 1, 1), B(0, -1, 4)$ and $C(k, 3, -2)$ are collinear, then $k =$ _____
Question 80 :
Let $\square PQRS$ be a quadrilateral. If $M$ and $N$ are midpoints of the sides $PQ$ and $RS$ respectively then $\overline {PS} + \overline {QR} =$
Question 82 :
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 83 :
<span>If two or more vectors are parallel to the same line, irrespective of their magnitudes and directions, then they are</span><br/>
Question 84 :
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 85 :
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 86 :
If the vectors $\bar { AB } =3\hat { i } +4\hat { k } $ and $\bar { AC } =5\hat { i } -2\hat j+4\hat k$<span> are the sides of a triangle ABC, then the length of the median through A is:</span>
Question 87 :
If $a$ and $b$ are two non collinear vectors and $x, y$ are two scalar such that $\vec ax + \vec b y = 0$ this implies that:
Question 88 :
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 89 :
$x$ 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 90 :
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 91 :
Line passing through $ (3,4,5) $ and $ (4,5,6) $ has direction ratios $ \ldots $
Question 93 :
If $\overrightarrow{a}$ and $\overrightarrow{b}$ two collinear vectors then which of the following are incorrect<br/>
Question 94 :
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/>
Question 96 :
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 97 :
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 98 :
If the position vectors of $P$ and $Q$ are $\overline{i}+2\overline{j}-7\overline{k}$ and $5\overline{i}-3\overline{j}+4\overline{k}$ respectively then the cosine of the angle between $\overline{PQ}$ and $z$-axis is<br/>
Question 99 :
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 100 :
The vector $b = 3j + 4k$ is to be written as the sum of a vector $b_{1}$ parallel to $a = i + j$ and a vector $b_{2}$ perpendicular to $a$. Then $b_{1}$ is equal to
