Let $\vec{a} = \alpha \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - \alpha \hat{k}$,where $\alpha > 0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $\vec{c} = -\hat{i} + 2 \hat{j} - 2 \hat{k}$ is $30$,then $\alpha$ is equal to:

$A$ tetrahedron has vertices $P(1, 2, 1)$,$Q(2, 1, 3)$,$R(-1, 1, 2)$ and $O(0, 0, 0)$. The angle between the faces $OPQ$ and $PQR$ is

Let the vectors $a, b, c$ and $d$ be such that $(a \times b) \times (c \times d) = 0$. Let $P_1$ and $P_2$ be planes determined by the pairs of vectors $(a, b)$ and $(c, d)$ respectively. Then the angle between $P_1$ and $P_2$ is:

$a = 3i - 5j$ and $b = 6i + 3j$ are two vectors and $c$ is a vector such that $c = a \times b$,then $|a|:|b|:|c|$ is

Let the lines $L_1: \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2}$ and $L_2: \frac{x - 2}{1} = \frac{y + 2}{2} = \frac{z - 3}{3}$. The unit vector perpendicular to both $L_1$ and $L_2$ is:

The moment of a force represented by $\overrightarrow{F} = i + 2j + 3k$ about the point $P(2, -1, 1)$ is given by the cross product of the position vector $\overrightarrow{r}$ and the force vector $\overrightarrow{F}$. If the origin is $O(0, 0, 0)$,then $\overrightarrow{r} = 2i - j + k$. Calculate the moment $\overrightarrow{M} = \overrightarrow{r} \times \overrightarrow{F}$.