If $\hat { u }$ and $\hat { v }$ are unit vectors and $\theta$ is the acute angle between them, then $2 \hat { u } \times 3 \hat { v }$ is a unit vector for
(1) exactly two values of $\theta$
(2) more than two values of $\theta$
(3) no value of $\theta$
(4) exactly one value of $\theta$

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c} = \hat{a} + 2\hat{b}$ and $\vec{d} = 5\hat{a} - 4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{2}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{4}$

If the vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a triangle $ABC$, then the length of the median through $A$ is:
(1) $\sqrt{33}$
(2) $\sqrt{45}$
(3) $\sqrt{18}$
(4) $\sqrt{72}$

If $[ \vec { a } \times \vec { b } \quad \vec { b } \times \vec { c } \quad \vec { c } \times \vec { a } ] = \lambda \left[ \vec { a } \quad \vec { b } \quad \vec { c } \right] ^ { 2 }$ then $\lambda$ is equal to
(1) 0
(2) 1
(3) 2
(4) 3

If $| \vec { a } | = 2 , | \vec { b } | = 3$ and $| \overrightarrow { 2 a } - \vec { b } | = 5$, then $| \overrightarrow { 2 a } + \vec { b } |$ equals:
(1) 5
(2) 7
(3) 17
(4) 1

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that no two of them are collinear and $( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }$. If $\theta$ is the angle between vectors $\vec { b }$ and $\vec { c }$, then a value of $\sin \theta$ is
(1) $\frac { - 2 \sqrt { 3 } } { 3 }$
(2) $\frac { 2 \sqrt { 2 } } { 3 }$
(3) $\frac { - \sqrt { 2 } } { 3 }$
(4) $\frac { 2 } { 3 }$

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 9$, then $|2\vec{a} + 5\vec{b} + 5\vec{c}|$ is:
(1) $3$
(2) $\sqrt{10}$
(3) $2$
(4) $\sqrt{5}$

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8 \hat { \mathrm { i } } - 6 \hat { \mathrm { j } }$ and $3 \hat { \mathrm { i } } + 4 \hat { \mathrm { j } } - 12 \widehat { \mathrm { k } }$, is:
(1) 20
(2) 65
(3) 52
(4) 26

Let $\vec { u }$ be a vector coplanar with the vectors $\vec { a } = 2 \hat { i } + 3 \hat { j } - \widehat { k }$ and $\vec { b } = \hat { j } + \widehat { k }$. If $\vec { u }$ is perpendicular to $\vec { a }$ and $\vec { u } \cdot \vec { b } = 24$, then $| \vec { u } | ^ { 2 }$ is equal to:
(1) 84
(2) 336
(3) 315
(4) 256

If $\vec { a } , \vec { b } , \vec { c }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \vec { c } = \overrightarrow { 0 }$, then $| \vec { a } \times \vec { c } |$ is equal to :
(1) $\frac { 1 } { 4 }$
(2) $\frac { 15 } { 16 }$
(3) $\frac { \sqrt { 15 } } { 4 }$
(4) $\frac { \sqrt { 15 } } { 16 }$

If $\vec { a } , \vec { b }$, and $\overrightarrow { \mathrm { c } }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \overrightarrow { \mathbf { c } } = \overrightarrow { 0 }$, then $| \vec { a } \times \overrightarrow { \mathbf { c } } |$ is equal to
(1) $\frac { 1 } { 4 }$
(2) $\frac { \sqrt { 15 } } { 4 }$
(3) $\frac { 15 } { 16 }$
(4) $\frac { \sqrt { 15 } } { 16 }$

Let $\vec { a } = 2 \hat { i } + \lambda _ { 1 } \hat { j } + 3 \hat { k } , \vec { b } = 4 \hat { i } + \left( 3 - \lambda _ { 2 } \right) \hat { j } + 6 \hat { k }$ and $\vec { c } = 3 \hat { i } + 6 \hat { j } + \left( \lambda _ { 3 } - 1 \right) \hat { k }$ be three vectors such that $\vec { b } = 2 \vec { a }$ and $\vec { a }$ is perpendicular to $\vec { c }$. Then a possible value of $\left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right)$ is
(1) $\left( - \frac { 1 } { 2 } , 4,0 \right)$
(2) $( 1,5,1 )$
(3) $\left( \frac { 1 } { 2 } , 4 , - 2 \right)$
(4) $( 1,3,1 )$

Let $\vec { a } = 3 \hat { i } + 2 \hat { j } + x \hat { k }$ and $\vec { b } = \hat { i } - \hat { j } + \hat { k }$, for some real $x$. Then the condition for $| \vec { a } \times \vec { b } | = r$ to follow
(1) $0 < r \leq \sqrt { \frac { 3 } { 2 } }$
(2) $r \geq 5 \sqrt { \frac { 3 } { 2 } }$
(3) $\sqrt { \frac { 3 } { 2 } } < r \leq 3 \sqrt { \frac { 3 } { 2 } }$
(4) $3 \sqrt { \frac { 3 } { 2 } } < r < 5 \sqrt { \frac { 3 } { 2 } }$

If a unit vector $\vec { a }$ makes angles $\frac { \pi } { 3 }$ with $\hat { i } , \frac { \pi } { 4 }$ with $\hat { j }$ and $\theta \in ( 0 , \pi )$ with $\widehat { k }$, then a value of $\theta$ is:
(1) $\frac { 5 \pi } { 6 }$
(2) $\frac { 5 \pi } { 12 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { 2 \pi } { 3 }$

Let the vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be such that $|\overrightarrow{\mathrm{a}}| = 2$, $|\overrightarrow{\mathrm{b}}| = 4$ and $|\overrightarrow{\mathrm{c}}| = 4$. If the projection of $\overrightarrow{\mathrm{b}}$ on $\overrightarrow{\mathrm{a}}$ is equal to the projection of $\overrightarrow{\mathrm{c}}$ on $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is perpendicular to $\overrightarrow{\mathrm{c}}$, then the value of $|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}|$ is ...

Assertion $A$ : If $A , B , C , D$ are four points on a semi-circular arc with a centre at $O$ such that $| \overrightarrow { A B } | = | \overrightarrow { B C } | = | \overrightarrow { C D } |$. Then, $\overrightarrow { A B } + \overrightarrow { A C } + \overrightarrow { A D } = 4 \overrightarrow { A O } + \overrightarrow { O B } + \overrightarrow { O C }$
Reason $R$ : Polygon law of vector addition yields $\overrightarrow { A B } + \overrightarrow { B C } + \overrightarrow { C D } + \overrightarrow { A D } = 2 \overrightarrow { A O }$
In the light of the above statements, choose the most appropriate answer from the options given below.
(1) $A$ is correct but $R$ is not correct.
(2) $A$ is not correct but $R$ is correct.
(3) Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$.
(4) Both $A$ and $R$ are correct but $R$ is not the correct explanation of $A$.

Let a vector $\alpha \hat { \mathrm { i } } + \beta \hat { \mathrm { j } }$ be obtained by rotating the vector $\sqrt { 3 } \hat { \mathrm { i } } + \hat { \mathrm { j } }$ by an angle $45 ^ { \circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $( \alpha , \beta ) , ( 0 , \beta )$ and $( 0,0 )$ is equal to
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $2 \sqrt { 2 }$

In a triangle $ABC$, if $| \overrightarrow { BC } | = 3 , | \overrightarrow { CA } | = 5$ and $| \overrightarrow { BA } | = 7$, then the projection of the vector $\overrightarrow { BA }$ on $\overrightarrow { BC }$ is equal to
(1) $\frac { 19 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 15 } { 2 }$

Let $O$ be the origin. Let $\overrightarrow { O P } = x \hat { i } + y \hat { j } - \widehat { k }$ and $\overrightarrow { O Q } = - \hat { i } + 2 \hat { j } + 3 x \hat { k } , x , y \in R , x > 0$, be such that $| \overrightarrow { P Q } | = \sqrt { 20 }$ and the vector $\overrightarrow { O P }$ is perpendicular to $\overrightarrow { O Q }$. If $\overrightarrow { O R } = 3 \hat { i } + \mathrm { z } \hat { j } - 7 \hat { k } , z \in R$, is coplanar with $\overrightarrow { O P }$ and $\overrightarrow { O Q }$, then the value of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ is equal to
(1) 7
(2) 9
(3) 2
(4) 1

Let $\vec { a }$ and $\vec { b }$ be two vectors such that $| 2 \vec { a } + 3 \vec { b } | = | 3 \vec { a } + \vec { b } |$ and the angle between $\vec { a }$ and $\vec { b }$ is $60 ^ { \circ }$. If $\frac { 1 } { 8 } \vec { a }$ is a unit vector, then $| \vec { b } |$ is equal to :
(1) 8
(2) 4
(3) 6
(4) 5

Let $\overrightarrow { \mathrm { a } } = \hat { \mathrm { i } } + \alpha \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { b } } = 3 \hat { \mathrm { i } } - \alpha \hat { \mathrm { j } } + \hat { \mathrm { k } }$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec { a }$ and $\vec { b }$ is $8 \sqrt { 3 }$ square units, then $\vec { a } \cdot \vec { b }$ is equal to $\underline{\hspace{1cm}}$.

Let $\vec { x }$ be a vector in the plane containing vectors $\vec { a } = 2 \hat { i } - \hat { j } + \hat { k }$ and $\vec { b } = \hat { i } + 2 \hat { j } - \hat { k }$. If the vector $\vec { x }$ is perpendicular to $( 3 \hat { i } + 2 \hat { j } - \widehat { k } )$ and its projection on $\vec { a }$ is $\frac { 17 \sqrt { 6 } } { 2 }$, then the value of $| \vec { x } | ^ { 2 }$ is equal to $\_\_\_\_$ .

For $p > 0$, a vector $\vec { v } _ { 2 } = 2 \hat { i } + ( p + 1 ) \hat { j }$ is obtained by rotating the vector $\vec { v } _ { 1 } = \sqrt { 3 } p \hat { i } + \hat { j }$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta = \frac { ( \alpha \sqrt { 3 } - 2 ) } { ( 4 \sqrt { 3 } + 3 ) }$, then the value of $\alpha$ is equal to $\underline{\hspace{1cm}}$.

Two vectors $\vec { A }$ and $\vec { B }$ have equal magnitudes. If magnitude of $\vec { A } + \vec { B }$ is equal to two times the magnitude of $\vec { A } - \vec { B }$, then the angle between $\vec { A }$ and $\vec { B }$ will be
(1) $\cos ^ { - 1 } \left( \frac { 3 } { 5 } \right)$
(2) $\cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(4) $\sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$

Two bodies of mass 1 kg and 3 kg have position vectors $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :
(1) $\hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$
(2) $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$
(3) $- 2 \hat { \mathrm { i } } + 2 \widehat { \mathrm { k } }$
(4) $- 2 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$