The adjacent sides of a parallelogram are $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$. Find its area.

Let a vector $\vec{a}$ have a magnitude $9$. Let a vector $\vec{b}$ be such that for every $(x, y) \in \mathbb{R} \times \mathbb{R} \setminus \{(0,0)\}$,the vector $(x \vec{a} + y \vec{b})$ is perpendicular to the vector $(6y \vec{a} - 18x \vec{b})$. Then the value of $|\vec{a} \times \vec{b}|$ is equal to:

If non-zero vectors $a$ and $b$ are perpendicular to each other,then what is the solution for $r \times a = b$?

The area of the parallelogram whose adjacent sides are $\hat{i}+\hat{k}$ and $2\hat{i}+\hat{j}+\hat{k}$ is

Let $a=2 \hat{i}-3 \hat{j}+4 \hat{k}$, $b=7 \hat{i}+2 \hat{j}-3 \hat{k}$, and $c=\hat{i}+\hat{j}+\hat{k}$. The vector $x$ such that $x \cdot c=60$ and $x$ is perpendicular to both $a$ and $b$ is:

The sine of the angle between the two vectors $3i + 2j - k$ and $12i + 5j - 5k$ will be