Let $|\vec{a}| = |\vec{b}| = 1$ and $|\vec{a} + \vec{b}| = \sqrt{3}$. If $\vec{c}$ is a vector satisfying the condition $\vec{c} - \vec{a} - 2\vec{b} = 3(\vec{a} \times \vec{b})$,then $\vec{c} \cdot \vec{b} = \dots$ (in $/2$)

If $U = [2, -3, 4]$,$X = [0, 2, 3]$,$V = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix}$,then $UV + XY = $?

$(\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = |\vec{a}|^2 + |\vec{b}|^2$ if and only if . . . . . . (where $\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}$).

If $|a| = 3, |b| = 1, |c| = 4$ and $a + b + c = 0,$ then $a \cdot b + b \cdot c + c \cdot a = $

Show that $|\vec{a}| \vec{b}+|\vec{b}| \vec{a}$ is perpendicular to $|\vec{a}| \vec{b}-|\vec{b}| \vec{a},$ for any two nonzero vectors $\vec{a}$ and $\vec{b}.$

Let $\overline{A}=2 \hat{i}+\hat{k}$,$\overline{B}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{C}=4 \hat{i}-3 \hat{j}+7 \hat{k}$. If a vector $\overline{R}$ satisfies $\overline{R} \times \overline{B}=\overline{C} \times \overline{B}$ and $\overline{R} \cdot \overline{A}=0$,then $\overline{R}$ is given by