Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\vec{b} \times \vec{a})-\vec{b}|^2$ is equal to

The area of a triangle with vertices $(1, 2, 0)$,$(1, 0, a)$,and $(0, 3, 1)$ is $\sqrt{6}$ sq. units. Then the values of '$a$' are:

If the position vectors of the vertices of $\triangle ABC$ are $3\hat{i}+4\hat{j}-\hat{k}$, $\hat{i}+3\hat{j}+\hat{k}$, and $5(\hat{i}+\hat{j}+\hat{k})$ respectively, then the magnitude of the altitude from $A$ onto the side $BC$ is

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$. If $|\vec{a} \times \vec{b}|=|\vec{a}|$,then the angle between the vectors $(\vec{a}+\vec{b}+(\vec{a} \times \vec{b}))$ and $\vec{a}$ is equal to

For any vector $x$, where $\hat{i}, \hat{j}, \hat{k}$ have their usual meanings, the value of $(x \times \hat{i})^{2} + (x \times \hat{j})^{2} + (x \times \hat{k})^{2}$ is equal to

If $\overline{a}=\hat{i}+4 \hat{j}+2 \hat{k}$,$\overline{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$,and $\overline{c}=2 \hat{i}-\hat{j}+4 \hat{k}$,then a vector $\bar{d}$ which is parallel to vector $\overline{a} \times \overline{b}$ and satisfies $\overline{c} \cdot \overline{d}=15$,is