If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular unit vectors, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

Let $\vec{a} = (\hat{i}+4\hat{j}+2\hat{k})$, $\vec{b} = (3\hat{i}-2\hat{j}+7\hat{k})$, $\vec{c} = (2\hat{i}-\hat{j}+4\hat{k})$. A vector $\vec{d}$, which is perpendicular to both $\vec{a}$ and $\vec{b}$, such that $\vec{c}.\vec{d} = 18$, is

If $\vec{a} = 2\hat{i} + m\hat{j} - n\hat{k}$ and $\vec{b} = l\hat{i} - 3\hat{j} + 4\hat{k}$ such that $\vec{a} = 2\vec{b}$ then the value of $14{l} + m + n$ is:

The projection of the vector $5\hat{i} + \hat{j} - 3\hat{k}$ on the vector $\hat{i} + 2\hat{j} - 3\hat{k}$ is

If $\vec{a}.\vec{b} = \vec{a}.\vec{c}$ and $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$, $\vec{a}\neq\vec{0}$, then

If $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $|2\vec{a} - \vec{b}| = 5$, then $|2\vec{a} + \vec{b}|$ equals to

A unit vector $\vec{a}$ makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with x-axis and y-axis respectively and an acute angle $\theta$ with z-axis, then the value of $\theta$ is