If $\vec{a}=2 \hat{i}+\hat{j}-3 \hat{k}$,$\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$,$\vec{c}=-\hat{i}+\hat{j}-4 \hat{k}$ and $\vec{d}=\hat{i}+\hat{j}+\hat{k}$,then $|(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})|=$

$A$ unit vector perpendicular to the vectors $4i - j + 3k$ and $-2i + j - 2k$ is

Let $\vec{a}=\hat{i}+\alpha \hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8 \sqrt{3}$ square units,then $\vec{a} \cdot \vec{b}$ is equal to ....... .

The unit vector perpendicular to $3i + 2j - k$ and $12i + 5j - 5k$ is

If the area of a parallelogram,whose diagonals are $\vec{d_1} = \hat{i} - \hat{j} + 2\hat{k}$ and $\vec{d_2} = 2\hat{i} + 3\hat{j} + \alpha\hat{k}$,is $\frac{\sqrt{93}}{2}$ sq. unit,then $\alpha = $

Let $\overline{a}, \overline{b}, \overline{c}$ be three vectors such that $|\overline{a}|=\sqrt{3}$,$|\overline{b}|=5$,$\overline{b} \cdot \overline{c}=10$ and the angle between $\overline{b}$ and $\overline{c}$ is $\frac{\pi}{3}$. If $\overline{a}$ is perpendicular to the vector $\overline{b} \times \overline{c}$,then $|\overline{a} \times(\overline{b} \times \overline{c})|$ is equal to