The volume of the tetrahedron having vertices $(1, -6, 10)$,$(-1, -3, 7)$,$(5, -1, \lambda)$ and $(7, -4, 7)$ is $11 \text{ cubic units}$. Then $\lambda = $

If $x$ is parallel to $y$ and $z$ where $x = 2i + j + \alpha k$,$y = \alpha i + k$ and $z = 5i - j$,then $\alpha$ is equal to

If $\vec{\alpha}$ is a unit vector,$\vec{\beta}=\hat{i}+\hat{j}-\hat{k}$,and $\vec{\gamma}=\hat{i}+\hat{k}$,then the maximum value of $[\vec{\alpha} \vec{\beta} \vec{\gamma}]$ is

If $\overline{a}, \overline{b}, \overline{c}$ are mutually perpendicular vectors having magnitudes $1, 2, 3$ respectively,then the value of $[\overline{a}+\overline{b}+\overline{c} \quad \overline{b}-\overline{a} \quad \overline{c}]$ is

The volume of a parallelepiped whose coterminous edges are represented by unit vectors $\hat{a}, \hat{b}, \hat{c}$ such that $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = \frac{1}{2}$ is:

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b}$,and $\vec{c} = \hat{j} - \hat{k}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 1$. If the length of the projection vector of the vector $\vec{b}$ on the vector $\vec{a} \times \vec{c}$ is $l$,then the value of $3l^{2}$ is equal to $.....$