If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0},$ find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}.$

If $a = 2i + 2j + 3k$,$b = -i + 2j + k$,and $c = 3i + j$,then $a + tb$ is perpendicular to $c$ if $t = $

Let $\bar{a} = 4\bar{i} + 3\bar{j}$ and $\bar{b}$ be two perpendicular vectors in the $XOY$-plane. $A$ vector $\bar{c}$ in the same plane and having projections $1$ and $2$ respectively on $\bar{a}$ and $\bar{b}$ is

If $a = t^2 \hat{i} + e^t \hat{j} + \hat{k}$ and $b = 2 \hat{i} + t^2 \hat{j} + \log t \hat{k}$,and $f(t) = a \cdot b$,then $f^{\prime}(1)$ is equal to

If $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ and $|\vec{a} \times \vec{b}| = |\vec{a} \cdot \vec{b}|$,then $\theta$ is equal to

If $\triangle ABC$ is right-angled at $A$,where $A \equiv (4, 2, x)$,$B \equiv (3, 1, 8)$,and $C \equiv (2, -1, 2)$,then the value of $x$ is