The value of $\lambda$ for which the vectors $2\lambda \hat{i} + \hat{j} - \hat{k}$ and $2\hat{j} + \hat{k}$ are perpendicular is:

If $a$ and $b$ are two unit vectors such that $a+2b$ and $5a - 4b$ are perpendicular to each other,then the angle between $a$ and $b$ is ............. $^o$

Let $a, b, c$ be unit vectors such that $a$ is perpendicular to the plane of $b$ and $c$. If the angle between $b$ and $c$ is $\frac{\pi}{3}$, then $|a + b + c| =$

In a parallelogram $ABCD$,$|\overline{AB}| = a$,$|\overline{AD}| = b$ and $|\overline{AC}| = c$,then the value of $\overline{DA} \cdot \overline{AB}$ is:

Three forces $i + 2j - 3k$,$2i + 3j + 4k$,and $i - j + k$ are acting on a particle at the point $(0, 1, 2)$. The magnitude of the moment of the forces about the point $(1, -2, 0)$ is

If $\vec{f}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{g}=2 \hat{i}-\hat{j}+3 \hat{k}$,then the projection vector of $\vec{f}$ on $\vec{g}$ is