Let $\vec{a} = \hat{i} + \alpha\hat{j} + \beta\hat{k}$, $\alpha, \beta \in R$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2 = 6$. If $\vec{a} \cdot \vec{b} = 3\sqrt{2}$, then the value of $(\alpha^2 + \beta^2)|\vec{a} \times \vec{b}|^2$ is equal to
(1) 90
(2) 75
(3) 95
(4) 85

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}| = 1$ and $|\vec{b} \times \vec{a}| = 2$. Then $|(\vec{b} \times \vec{a}) - \vec{b}|^2$ is equal to
(1) 3
(2) 5
(3) 1
(4) 4

Let $\vec { a } = 9 \hat { i } - 13 \hat { j } + 25 \hat { k } , \vec { b } = 3 \hat { i } + 7 \hat { j } - 13 \hat { k }$ and $\vec { c } = 17 \hat { i } - 2 \hat { j } + \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = ( \vec { b } + \vec { c } ) \times \vec { a }$ and $\vec { r } \cdot ( \vec { b } - \vec { c } ) = 0$, then $\frac { | 593 \vec { r } + 67 \vec { a } | ^ { 2 } } { ( 593 ) ^ { 2 } }$ is equal to $\_\_\_\_$

Let $\vec { a } = \hat { \mathrm { i } } + \hat { \mathrm { j } } + \hat { \mathrm { k } } , \overrightarrow { \mathrm { b } } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { d } } = \vec { a } \times \overrightarrow { \mathrm { b } }$. If $\overrightarrow { \mathrm { c } }$ is a vector such that $\vec { a } \cdot \overrightarrow { \mathrm { c } } = | \overrightarrow { \mathrm { c } } | , | \overrightarrow { \mathrm { c } } - 2 \vec { a } | ^ { 2 } = 8$ and the angle between $\overrightarrow { \mathrm { d } }$ and $\overrightarrow { \mathrm { c } }$ is $\frac { \pi } { 4 }$, then $| 10 - 3 \overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { c } } | + | \overrightarrow { \mathrm { d } } \times \overrightarrow { \mathrm { c } } | ^ { 2 }$ is equal to

If the components of $\overrightarrow { \mathrm { a } } = \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k }$ along and perpendicular to $\overrightarrow { \mathrm { b } } = 3 \hat { i } + \hat { j } - \hat { k }$ respectively, are $\frac { 16 } { 11 } ( 3 \hat { i } + \hat { j } - \hat { k } )$ and $\frac { 1 } { 11 } ( - 4 \hat { i } - 5 \hat { j } - 17 \hat { k } )$, then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to :
(1) 26
(2) 18
(3) 23
(4) 16

It is known that a right triangle $\triangle A B C$ has side lengths $\overline { A B } = \sqrt { 7 }$, $\overline { A C } = \sqrt { 3 }$, $\overline { B C } = 2$. If isosceles triangles $\triangle M A B$ and $\triangle N A C$ with vertex angles equal to $120 ^ { \circ }$ are constructed outside $\triangle A B C$ using $\overline { A B }$ and $\overline { A C }$ as bases respectively, then $\overline { M N } ^ { 2 } =$ . (Express as a fraction in lowest terms)