The area of the parallelogram for which the vectors $\hat{i}+\hat{j}+2 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$ are adjacent sides is equal to

If vectors $a, b,$ and $c$ represent the sides $BC, CA,$ and $AB$ of a triangle $ABC$ respectively,then which of the following is true?

Let $p, q$ and $r$ be vectors such that $r \neq 0$,$p \times q = r$,and $q \times p = r$. Then which of the following is true?
$(i)$ $p, q, r$ are pair-wise orthogonal vectors
(ii) $|q| = |r| = |p|$

For a triangle $ABC$,let $\vec{p}=\vec{BC}$,$\vec{q}=\vec{CA}$ and $\vec{r}=\vec{BA}$. If $|\vec{p}|=2\sqrt{3}$,$|\vec{q}|=2$ and $\cos \theta = \frac{1}{\sqrt{3}}$ where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$,then $|\vec{p} \times (\vec{q}-3\vec{r})|^{2}+3|\vec{r}|^{2}$ is equal to:

Find the unit vector which is perpendicular to the vector $5i + 2j + 6k$ and coplanar with the vectors $2i + j + k$ and $i - j + k$.

Let $p, q, r$ be three mutually perpendicular vectors of the same magnitude. If a vector $x$ satisfies the equation $p \times \{(x - q) \times p\} + q \times \{(x - r) \times q\} + r \times \{(x - p) \times r\} = 0$,then $x$ is given by