If the vectors $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$,$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$,and $\vec{c} = \lambda\hat{i} + \hat{j} + \mu\hat{k}$ are mutually perpendicular,then $(\lambda, \mu) = .......$

If $\hat{a}, \hat{b}, \hat{c}$ are unit vectors,then the least value of $|\hat{a}+\hat{b}|^2+|\hat{b}+\hat{c}|^2+|\hat{c}+\hat{a}|^2$ will be-

Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$ with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36 \cos ^{2} 2 \theta$ is equal to $.....$

Let the vectors $\vec{a} = -\hat{i} + \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + \hat{k}$. For some $\lambda, \mu \in \mathbb{R}$, let $\vec{c} = \lambda \vec{a} + \mu \vec{b}$. If $\vec{c} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 10$ and $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = -2$, then $|\vec{c}|^2$ is equal to:

Let $a = \sin^2 x \hat{i} + \cos^2 x \hat{j} + \hat{k}$, where $x \in R$. If the pairs of vectors $(a, \hat{i})$, $(a, \hat{j})$, and $(a, \hat{k})$ are adjacent sides of $3$ distinct parallelograms and $A$ is the sum of the squares of the areas of these parallelograms, then $A$ lies in the interval

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - 3\hat{j} + 2\hat{k}$ and $\vec{c} = 3\hat{i} - 2\hat{k}$. If a vector $\vec{p}$ satisfies the conditions $\vec{p} \cdot \vec{c} = 0$ and $\vec{p} \times \vec{a} = \vec{b} \times \vec{a}$, then the value of $|\vec{p}|$ is: