If $a, b, c$ are three non-coplanar vectors and $p, q, r$ are defined by the relations $p = \frac{b \times c}{[a, b, c]}, q = \frac{c \times a}{[a, b, c]}, r = \frac{a \times b}{[a, b, c]}$,then $(a+b) \cdot p + (b+c) \cdot q + (c+a) \cdot r =$

Let $O$ be the origin. Let $\overline{OP} = x\hat{i} + y\hat{j} - \hat{k}$ and $\overline{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$,where $x, y \in \mathbb{R}$ and $x > 0$,be such that $|\overline{PQ}| = \sqrt{20}$ and the vector $\overline{OP}$ is perpendicular to $\overline{OQ}$. If $\overline{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$,where $z \in \mathbb{R}$,is coplanar with $\overline{OP}$ and $\overline{OQ}$,then the value of $x^2 + y^2 + z^2$ is equal to ...... .

The volume of the tetrahedron,whose vertices are given by the vectors $-i + j + k$,$i - j + k$,and $i + j - k$ with reference to the fourth vertex as the origin,is

If $a = i - j + k$,$b = i + 2j - k$ and $c = 3i + pj + 5k$ are coplanar,then the value of $p$ is:

If $\bar{a}=\hat{i}+5 \hat{k}, \bar{b}=2 \hat{i}+3 \hat{k}, \bar{c}=4 \hat{i}-\hat{j}+2 \hat{k}$ and $\bar{d}=\hat{i}-\hat{j}$,then $(\bar{c}-\bar{a}) \cdot(\bar{b} \times \bar{d})=$

Let $\alpha \in \mathbb{R}$ and the three vectors $\vec{a} = \alpha \hat{i} + \hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + \hat{j} - \alpha \hat{k}$,and $\vec{c} = \alpha \hat{i} - 2\hat{j} + 3\hat{k}$. Then the set $S = \{ \alpha : \vec{a}, \vec{b}, \text{ and } \vec{c} \text{ are coplanar} \}$