The volume of the tetrahedron whose coterminus edges are represented by $\bar{a}=-12 \hat{i}+p \hat{k}$,$\bar{b}=3 \hat{j}-\hat{k}$,and $\bar{c}=2 \hat{i}+\hat{j}-15 \hat{k}$ is $570$ cubic units. Then,$p=$

If the volume of a parallelepiped with coterminous edges $4 \hat{i} + 5 \hat{j} + \hat{k}$, $-\hat{j} + \hat{k}$, and $3 \hat{i} + 9 \hat{j} + p \hat{k}$ is $34$ cubic units, then $p$ is equal to:

Consider the vectors $\vec{x}=\hat{i}+2\hat{j}+3\hat{k}$,$\vec{y}=2\hat{i}+3\hat{j}+\hat{k}$,and $\vec{z}=3\hat{i}+\hat{j}+2\hat{k}$. For two distinct positive real numbers $\alpha$ and $\beta$,define $\vec{X}=\alpha\vec{x}+\beta\vec{y}-\vec{z}$,$\vec{Y}=\alpha\vec{y}+\beta\vec{z}-\vec{x}$,and $\vec{Z}=\alpha\vec{z}+\beta\vec{x}-\vec{y}$. If the vectors $\vec{X}, \vec{Y}$,and $\vec{Z}$ lie in a plane,the value of $\alpha+\beta-3$ is $....$.

If a vector $\alpha$ lies in the plane of $\beta$ and $\gamma$,then which of the following is correct?

Value of $\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{i} \cdot(\hat{j} \times \hat{j})+\hat{k} \cdot(\hat{j} \times \hat{i})+\hat{i} \cdot(\hat{k} \times \hat{j})$ is . . . . . . .

If $a, b, c$ are three non-coplanar vectors and $d$ is any unit vector, then $|(a \cdot d)(b \times c) + (b \cdot d)(c \times a) + (c \cdot d)(a \times b)| = $