If the vectors $i + 3j$,$5k$,and $Pi - j$ are coplanar,find the value of $P$.

If $a = -3i + 7j + 5k$,$b = -3i + 7j - 3k$,and $c = 7i - 5j - 3k$ are the three coterminous edges of a parallelepiped,then its volume is

If $\vec{p}$ and $\vec{q}$ are unequal unit vectors such that $(\vec{p} - \vec{q}) \cdot ((2\vec{q} + \vec{p}) \times (3\vec{p} - \vec{q})) = |\vec{p} + \vec{q}|$,then the angle between $\vec{p}$ and $\vec{q}$ is:

For three vectors $a, b, c$,the value of $[a \times b, b \times c, c \times a]$ is equal to:

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and the points $P_1 = \lambda \vec{a}+3 \vec{b}-\vec{c}$,$P_2 = \vec{a}-\lambda \vec{b}+3 \vec{c}$,$P_3 = 3 \vec{a}+4 \vec{b}-\lambda \vec{c}$,and $P_4 = \vec{a}-6 \vec{b}+6 \vec{c}$ are coplanar,then one of the values of $\lambda$ is

If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $