Let $\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$, $\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$ and $\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k$ be three vectors such that, $\left| {\overrightarrow b \times \overrightarrow c } \right| = 5\sqrt 3$ and ${\overrightarrow a }$ is perpendicular to ${\overrightarrow b }$. Then the greatest amongst the values of ${\left| {\overrightarrow a } \right|^2}$ is _____________.

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}…