If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to both $\vec{b}$ and $\vec{c}$,and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2 \pi}{3}$,then $|\vec{a}+3 \vec{b}-4 \vec{c}|^2=$

Let $\theta$ denote the angle between vectors $\vec{a}$ and $\vec{b}$. If $\vec{a}=2 \hat{i}+3 \hat{j}+6 \hat{k}$,$\vec{a} \cdot \vec{b}=4$ and $\theta=\cos ^{-1}\left(\frac{4}{21}\right)$,then $\vec{a}+\vec{b}$ is:

Let $\overrightarrow{a} = 2\hat{i} + \hat{j} + \hat{k}$,and $\overrightarrow{b}$ and $\overrightarrow{c}$ be two nonzero vectors such that $|\vec{a} + \vec{b} + \vec{c}| = |\vec{a} + \vec{b} - \vec{c}|$ and $\vec{b} \cdot \vec{c} = 0$. Consider the following two statements:
$(A)$ $|\overrightarrow{a} + \lambda \overrightarrow{c}| \geq |\overrightarrow{a}|$ for all $\lambda \in R$.
$(B)$ $\overrightarrow{a}$ and $\overrightarrow{c}$ are always parallel.

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors and $\vec{a}+\vec{b}+\vec{c}=\vec{0}$,then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is . . . . . . . (in $/2$)

If $\overline{a}, \overline{b}, \overline{c}$ are unit vectors and $\theta$ is the angle between $\overline{a}$ and $\overline{c}$ and $\overline{a}+2 \overline{b}+2 \overline{c}=\overline{0}$,then $|\overline{a} \times \overline{c}|=$

If the vector $a + b$ makes equal angles with vectors $a$ and $b$,then: