If $2 \hat{i}-\hat{j}+3 \hat{k}$,$-12 \hat{i}-\hat{j}-3 \hat{k}$,$-\hat{i}+2 \hat{j}-4 \hat{k}$ and $\lambda \hat{i}+2 \hat{j}-\hat{k}$ are the position vectors of four coplanar points,then $\lambda=$

If $x, y$ and $z$ are non-zero real numbers and $\vec{a}=x \hat{i}+2 \hat{j}, \vec{b}=y \hat{j}+3 \hat{k}$ and $\vec{c}=x \hat{i}+y \hat{j}+z \hat{k}$ are such that $\vec{a} \times \vec{b}=z \hat{i}-3 \hat{j}+\hat{k}$,then $[\vec{a} \vec{b} \vec{c}]$ equals to

If $x \cdot a = 0, x \cdot b = 0$ and $x \cdot c = 0$ for some non-zero vector $x$,then the true statement is

Let the vectors $\overrightarrow{u}_1 = \hat{i} + \hat{j} + a\hat{k}$,$\overrightarrow{u}_2 = \hat{i} + b\hat{j} + \hat{k}$ and $\overrightarrow{u}_3 = c\hat{i} + \hat{j} + \hat{k}$ be coplanar. If the vectors $\overrightarrow{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$,$\overrightarrow{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$ and $\overrightarrow{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$ are also coplanar,then $6(a+b+c)$ is equal to $..............$.

If the three coterminous edges of a parallelepiped are represented by the vectors $(a - b)$,$(b - c)$,and $(c - a)$,find its volume.

Let the vectors $\vec{u} = (2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}$,$\vec{v} = (1+b) \hat{i}+2 b \hat{j}-b \hat{k}$,and $\vec{w} = (2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}$ where $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true?