If $(1,5,35), (7,5,5), (1, \lambda, 7)$ and $(2 \lambda, 1, 2)$ are coplanar,then the sum of all possible values of $\lambda$ is

Evaluate: $\vec{a} \cdot \{(\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})\}$

If $\bar{u}=\hat{\imath}-2 \hat{\jmath}+\hat{k}, \bar{v}=3 \hat{\imath}+\hat{k}$ and $\bar{w}=\hat{\jmath}-\hat{k}$,then the volume of the parallelepiped with $\bar{u} \times \bar{v}, \bar{u}+\bar{w}$ and $\bar{v}+\bar{w}$ as coterminus edges is

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,$p=$

Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$,$\vec{b} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{c} = x\vec{a} + y\vec{b} + z(\vec{a} \times \vec{b})$,then the value of $x + y + z$ is:

Consider the set of eight vectors $V=\{a \hat{i}+b \hat{j}+c \hat{k}: a, b, c \in\{-1,1\}\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is