If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are unit vectors such that $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1$ and $\vec{a} \cdot \vec{c} = \frac{1}{2}$,then :-

The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is

Given the following simultaneous equations for vectors $x$ and $y$:
$(i) x + y = a$
$(ii) x \times y = b$
$(iii) x \cdot a = 1$
Then $x = ?, y = ?$

If the vectors $\overline{a} = c(\log_7 x) \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $\overline{b} = (\log_7 x) \hat{i} + 3c(\log_7 x) \hat{j} - 4 \hat{k}$ make an obtuse angle for any $x > 0$,then $c$ belongs to

If with reference to the right-handed system of mutually perpendicular unit vectors $\hat{i}, \hat{j}$ and $\hat{k}$,$\vec{\alpha} = 3\hat{i} - \hat{j}$ and $\vec{\beta} = 2\hat{i} + \hat{j} - 3\hat{k}$,then express $\vec{\beta}$ in the form $\vec{\beta} = \vec{\beta}_{1} + \vec{\beta}_{2}$,where $\vec{\beta}_{1}$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_{2}$ is perpendicular to $\vec{\alpha}$.

If $a=2 \hat{i}+\hat{k}$,$b=\hat{i}+\hat{j}+\hat{k}$,and $c=4 \hat{i}-3 \hat{j}+7 \hat{k}$,then the vector $r$ satisfying $r \times b=c \times b$ and $r \cdot a=0$ is