If $a$ is perpendicular to $b$ and $c$,$|a| = 2$,$|b| = 3$,$|c| = 4$ and the angle between $b$ and $c$ is $\frac{2\pi}{3}$,then $[a \; b \; c]$ is equal to (in $\sqrt{3}$):

Let $a = \hat{i} - 2\hat{j} + 3\hat{k}$ and $b = 2\hat{i} + \hat{j} + \hat{k}$. If $c$ is a unit vector such that $[a \ b \ c]$ is maximum, then $c =$

If the vectors $2 \hat{i}-\hat{j}-\hat{k}$,$\hat{i}+2 \hat{j}-3 \hat{k}$,and $3 \hat{i}+\lambda \hat{j}+5 \hat{k}$ are coplanar,then the value of $\lambda$ is

If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + aj + 5k$ are coplanar,find the value of $a$.

Let $\overrightarrow{a}=\hat{i}-2 \hat{j}+3 \hat{k}$, $\overrightarrow{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}$. If $\overrightarrow{c}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\lambda$ is equal to

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