If $\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k$,

$\overrightarrow b = - \beta \widehat i - \alpha \widehat j - \widehat k$ and

$\overrightarrow c = \widehat i - 2\widehat j - \widehat k$

such that $\overrightarrow a \,.\,\overrightarrow b = 1$ and $\overrightarrow b \,.\,\overrightarrow c = - 3$, then $${1 \over 3}\left( {\left( {\overrightarrow a \times \overrightarrow b } \right)\,.\,\overrightarrow c } \right)$$ is equal to _____________.

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$ and…

If $\left( {\overrightarrow a + 3\overrightarrow b } \right)$ is perpendicular to $\left( {7\overrightarrow a - 5\overrightarrow b }…

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha…

Let $\overrightarrow a$ = $\widehat i$ + 2$\widehat j$ $-$ 3$\widehat k$ and $\overrightarrow b = 2\widehat i$ $-$ 3$\widehat j$ +…

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$, and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a}…