If the vectors $\vec P = a\hat i + a\hat j + 3\hat k$ and $\vec Q = a\hat i - 2\hat j - \hat k$ are perpendicular to each other,then the positive value of $a$ is

Two position vectors are given by $\overrightarrow{r_1} = (1, 1, 1)$ and $\overrightarrow{r_2} = (1, -1, 1)$. Find the unit vector in the direction of $\overrightarrow{r_1} \times \overrightarrow{r_2}$.

For what value of $x$ will the two vectors $\vec{A} = 2\hat{i} + 2\hat{j} - x\hat{k}$ and $\vec{B} = 2\hat{i} - \hat{j} - 3\hat{k}$ be perpendicular to each other?

The angle between $(\overrightarrow{A} - \overrightarrow{B})$ and $(\overrightarrow{A} \times \overrightarrow{B})$ is $(\overrightarrow{A} \neq \overrightarrow{B})$. (in $^{\circ}$)

Let $\vec A = (\hat i + \hat j)$ and $\vec B = (2\hat i - \hat j)$. The magnitude of a coplanar vector $\vec C$ such that $\vec A \cdot \vec C = \vec B \cdot \vec C = \vec A \cdot \vec B$ is given by

If a vector $A=2i+2j+3k$ and $B=3i+6j+nk$ are perpendicular to each other,then the value of $n$ is