Let $\vec{a}$ be a unit vector and $\vec{b}$ be a nonzero vector not parallel to $\vec{a}$. The angles of the triangle,two of whose sides are represented by $\sqrt{3}(\vec{a} \times \vec{b})$ and $\vec{b} - (\vec{a} \cdot \vec{b})\vec{a}$,are

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

If $\overrightarrow{A}=i-2j-3k,\,\overrightarrow{B}=2i+j-k,\,\overrightarrow{C}=i+3j-2k$,then $(\overrightarrow A \times \overrightarrow B ) \times \overrightarrow C $ is

If the area of the triangle with vertices $(1, 2, 0)$,$(1, 0, 2)$ and $(0, x, 1)$ is $\sqrt{6}$ square units,then the value of $x$ is

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 $|(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})|=$

If $a, b, c, d$ are coplanar vectors,then $(a \times b) \times (c \times d)$ is equal to