If $\vec { A } = 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } - \hat { \mathrm { k } }$ m and $\vec { B } = \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ m. The magnitude of component of vector $\vec { A }$ along vector $\vec { B }$ will be $\_\_\_\_$ m.

If the projection of $2\hat{i} + 4\hat{j} - 2\hat{k}$ on $\hat{i} + 2\hat{j} + \alpha\hat{k}$ is zero. Then, the value of $\alpha$ will be

Let $a$ and $b$ be two unit vectors such that $| ( a + b ) + 2 ( a \times b ) | = 2$. If $\theta \in ( 0 , \pi )$ is the angle between $\hat { \mathrm { a } }$ and $\widehat { \mathrm { b } }$, then among the statements: $( S 1 ) : 2 | \widehat { a } \times \hat { b } | = | \widehat { a } - \hat { b } |$ $( S 2 )$ : The projection of $\widehat { a }$ on $( \widehat { a } + \widehat { b } )$ is $\frac { 1 } { 2 }$
(1) Only $( S 1 )$ is true.
(2) Only $( S 2 )$ is true.
(3) Both $( S 1 )$ and $( S 2 )$ are true.
(4) Both $( S 1 )$ and $( S 2 )$ are false.

Let $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a} \times \vec{b} = 2\hat{i} - \hat{k}$ and $\vec{a} \cdot \vec{b} = 3$. Then the projection of $\vec{b}$ on the vector $\vec{a} - \vec{b}$ is:
(1) $\frac{2}{\sqrt{21}}$
(2) $2\sqrt{\frac{3}{7}}$
(3) $\frac{2}{3}\sqrt{\frac{7}{3}}$
(4) $\frac{2}{3}$

If $\vec { a } \cdot \vec { b } = 1 , \vec { b } \cdot \vec { c } = 2$ and $\vec { c } \cdot \vec { a } = 3$, then the value of $[ \vec { a } \times ( \vec { b } \times \vec { c } ) \quad \vec { b } \times ( \vec { c } \times \vec { a } ) \quad \vec { c } \times ( \vec { b } \times \vec { a } ) ]$ is
(1) 0
(2) $- 6 \vec { a } \cdot ( \vec { b } \times \vec { c } )$
(3) $12 \vec { c } \cdot ( \vec { a } \times \vec { b } )$
(4) $- 12 \vec { b } \cdot ( \vec { c } \times \vec { a } )$

Let $\vec { a } = \alpha \hat { i } + \hat { j } - \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \alpha \hat { k } , \alpha > 0$. If the projection of $\vec { a } \times \vec { b }$ on the vector $- \hat { i } + 2 \hat { j } - 2 \hat { k }$ is 30, then $\alpha$ is equal to
(1) $\frac { 15 } { 2 }$
(2) 8
(3) $\frac { 13 } { 2 }$
(4) 7

Let $\vec { a } = \alpha \hat { i } + \hat { j } + \beta \hat { k }$ and $\vec { b } = 3 \hat { i } - 5 \hat { j } + 4 \hat { k }$ be two vectors, such that $\vec { a } \times \vec { b } = - \hat { i } + 9 \hat { i } + 12 \widehat { k }$. Then the projection of $\vec { b } - 2 \vec { a }$ on $\vec { b } + \vec { a }$ is equal to
(1) 2
(2) $\frac { 39 } { 5 }$
(3) 9
(4) $\frac { 46 } { 5 }$

Let $S$ be the set of all $a \in R$ for which the angle between the vectors $\vec { u } = a \left( \log _ { e } b \right) \hat { i } - 6 \hat { j } + 3 \hat { k }$ and $\vec { v } = \left( \log _ { e } b \right) \hat { i } + 2 \hat { j } + 2 a \left( \log _ { e } b \right) \hat { k } , ( b > 1 )$ is acute. Then $S$ is equal to
(1) $\left( - \infty , - \frac { 4 } { 3 } \right)$
(2) $\Phi$
(3) $\left( - \frac { 4 } { 3 } , 0 \right)$
(4) $\left( \frac { 12 } { 7 } , \infty \right)$

Let $\vec { a } = \hat { i } + \hat { j } + 2 \widehat { k } , \vec { b } = 2 \hat { i } - 3 \hat { j } + \widehat { k }$ and $\vec { c } = \hat { i } - \hat { j } + \widehat { k }$ be the three given vectors. Let $\vec { v }$ be a vector in the plane of $\vec { a }$ and $\vec { b }$ whose projection on $\vec { c }$ is $\frac { 2 } { \sqrt { 3 } }$. If $\vec { v } \cdot \hat { j } = 7$, then $\vec { v } \cdot ( \hat { i } + \hat { k } )$ is equal to
(1) 6
(2) 7
(3) 8
(4) 9

Let $\vec{a}, \vec{b}, \vec{c}$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = 168$, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to
(1) 10
(2) 14
(3) 16
(4) 18

Let $\vec { a } = 2 \hat { i } - \hat { j } + 5 \hat { k }$ and $\vec { b } = \alpha \hat { i } + \beta \hat { j } + 2 \widehat { k }$. If $( ( \vec { a } \times \vec { b } ) \times \hat { i } ) \cdot \widehat { k } = \frac { 23 } { 2 }$, then $| \vec { b } \times 2 \hat { j } |$ is equal to
(1) 4
(2) 5
(3) $\sqrt { 21 }$
(4) $\sqrt { 17 }$

Let $\vec { a } , \vec { b } , \vec { c }$ be three non-coplanar vectors such that $\vec { a } \times \vec { b } = \overrightarrow { 4 c } , \vec { b } \times \vec { c } = 9 \vec { a }$ and $\vec { c } \times \vec { a } = \alpha \vec { b } , \alpha > 0$. If $| \vec { a } | + | \vec { b } | + | \vec { c } | = 36$, then $\alpha$ is equal to $\_\_\_\_$ .

When vector $\vec { A } = 2 \hat { i } + 3 \hat { j } + 2 \widehat { k }$ is subtracted from vector $\vec { B }$, it gives a vector equal to $2 \hat { j }$. Then the magnitude of vector $\vec { B }$ will be:
(1) $\sqrt { 5 }$
(2) 3
(3) $\sqrt { 6 }$
(4) $\sqrt { 33 }$

Vectors $a \hat { i } + b \hat { j } + \hat { k }$ and $2 \hat { i } - 3 \hat { j } + 4 \hat { k }$ are perpendicular to each other when $3 a + 2 b = 7$, the ratio of $a$ to $b$ is $\frac { x } { 2 }$. The value of $x$ is $\_\_\_\_$ .

An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\overrightarrow{OP} = \vec{u}$, $\overrightarrow{OR} = \vec{v}$ and $\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$, then $\alpha$, $\beta^2$ are the roots of the equation
(1) $x^2 + x - 2 = 0$
(2) $x^2 - x - 2 = 0$
(3) $3x^2 - 2x - 1 = 0$
(4) $3x^2 + 2x - 1 = 0$

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i} - 2\hat{j} + 3\hat{k}$. If the position vectors of the points $A$, $B$ and $C$ are $-2\hat{i} + \hat{j} - 3\hat{k}$, $2\hat{i} + 4\hat{j} - 2\hat{k}$ and $-4\hat{i} + 2\hat{j} - \hat{k}$ respectively, then the projection of the vector $\overrightarrow{OP}$ on a vector perpendicular to the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is
(1) 3
(2) $\frac{8}{3}$
(3) $\frac{7}{3}$
(4) $\frac{10}{3}$

Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $( \overrightarrow { AB } - \overrightarrow { BC } ) + ( \overrightarrow { AD } - \overrightarrow { DC } ) = k \overrightarrow { FE }$, then $k$ is equal to
(1) 4
(2) $- 2$
(3) 2
(4) $- 4$

Let $a , b , c$ be three distinct real numbers, none equal to one. If the vectors $a \hat { i } + \hat { j } + \widehat { k } , \hat { i } + b \hat { j } + \widehat { k }$ and $\hat { i } + \hat { j } + c \hat { k }$ are coplanar, then $\frac { 1 } { 1 - a } + \frac { 1 } { 1 - b } + \frac { 1 } { 1 - c }$ is equal to
(1) 2
(2) - 1
(3) - 2
(4) 1

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non zero vectors such that $\vec { b } \cdot \vec { c } = 0$ and $\vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { \vec { b } - \vec { c } } { 2 }$. If $\vec { d }$ be a vector such that $\overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { d } } = \overrightarrow { \mathrm { a } } \cdot \overrightarrow { \mathrm { b } }$, then $( \overrightarrow { \mathrm { a } } \times \overrightarrow { \mathrm { b } } ) \cdot ( \overrightarrow { \mathrm { c } } \times \overrightarrow { \mathrm { d } } )$ is equal to
(1) $\frac { 3 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $- \frac { 1 } { 4 }$
(4) $\frac { 1 } { 4 }$

If $\vec { a } = \hat { i } + 2 \widehat { k } , \vec { b } = \hat { i } + \hat { j } + \widehat { k } , \vec { c } = 7 \hat { i } - 3 \hat { j } + 4 \widehat { k } , \vec { r } \times \vec { b } + \vec { b } \times \vec { c } = \overrightarrow { 0 }$ and $\vec { r } \cdot \vec { a } = 0$ then $\vec { r } \cdot \vec { c }$ is equal to: (1) 34 (2) 12 (3) 36 (4) 30

Let $\lambda \in \mathbb { Z } , \vec { a } = \lambda \hat { i } + \hat { j } - \widehat { k }$ and $\vec { b } = 3 \hat { i } - \hat { j } + 2 \widehat { k }$. Let $\vec { c }$ be a vector such that $( \vec { a } + \vec { b } + \vec { c } ) \times \vec { c } = \overrightarrow { 0 } , \vec { a } \cdot \vec { c } = - 17$ and $\vec { b } \cdot \vec { c } = - 20$. Then $| \vec { c } \times ( \lambda \hat { i } + \hat { j } + \hat { k } ) | ^ { 2 }$ is equal to
(1) 46
(2) 53
(3) 62
(4) 49

The vector $\vec { a } = - \hat { i } + 2 \hat { j } + \hat { k }$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec { b }$. Then the projection of $3 \vec { a } + \sqrt { 2 } \vec { b }$ on $\vec { c } = 5 \hat { i } + 4 \hat { j } + 3 \hat { k }$ is
(1) $3 \sqrt { 2 }$
(2) 1
(3) $\sqrt { 6 }$
(4) $2 \sqrt { 3 }$

Let $\vec { a } = 4 \hat { i } + 3 \hat { j }$ and $\vec { b } = 3 \hat { i } - 4 \hat { j } + 5 \hat { k }$ and $\overrightarrow { \mathrm { c } }$ is a vector such that $\vec { c } \cdot ( \vec { a } \times \vec { b } ) + 25 = 0 , \vec { c } \cdot ( \hat { i } + \hat { j } + \hat { k } ) = 4$ and projection of $\vec { c }$ on $\overrightarrow { \mathrm { a } }$ is 1 , then the projection of $\vec { c }$ on $\vec { b }$ equals: (1) $\frac { 5 } { \sqrt { 2 } }$ (2) $\frac { 1 } { 5 }$ (3) $\frac { 1 } { \sqrt { 2 } }$ (4) $\frac { 3 } { \sqrt { 2 } }$

Let $\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}| = \alpha > 0$. If the minimum value of the scalar triple product $[\vec{u}\, \vec{v}\, \vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to $\_\_\_\_$.

$A(2, 6, 2)$, $B(-4, 0, \lambda)$, $C(2, 3, -1)$ and $D(4, 5, 0)$, $\lambda \leq 5$ are the vertices of a quadrilateral $ABCD$. If its area is 18 square units, then $5 - 6\lambda$ is equal to $\_\_\_\_$.