The vectors $i + 2j + 3k$,$\lambda i + 4j + 7k$,and $-3i - 2j - 5k$ are collinear if $\lambda$ equals:

The volume of the tetrahedron with $\hat{i}-\lambda \hat{j}+\hat{k}$,$\lambda \hat{i}-\hat{j}-\hat{k}$ and $\hat{i}+\hat{j}+\lambda \hat{k}$ as coterminous edges is $2$. If $\lambda$ is an integer,then $|\lambda \hat{i}-3 \lambda \hat{j}+3 \hat{k}|=$

If three non-zero vectors are $a = a_1 i + a_2 j + a_3 k,$ $b = b_1 i + b_2 j + b_3 k$ and $c = c_1 i + c_2 j + c_3 k.$ If $c$ is the unit vector perpendicular to the vectors $a$ and $b$ and the angle between $a$ and $b$ is $\frac{\pi}{6},$ then $\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|^2$ is equal to

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

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelepiped of volume $V$. Then the volume of the parallelepiped,whose coterminous edges are represented by $\vec{a}, \vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to $..........\,V$.

$a \cdot (b \times c)$ is equal to