If $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{c} = r\hat{i} + \hat{j} + (2r - 1)\hat{k}$ are three vectors such that $\vec{c}$ is parallel to the plane of $\vec{a}$ and $\vec{b}$,then $r$ is equal to

The volume of a parallelepiped,whose coterminous edges are given by $\bar{u}=\hat{i}+\hat{j}+\lambda \hat{k}$,$\bar{v}=\hat{i}+\hat{j}+3 \hat{k}$,and $\bar{w}=2 \hat{i}+\hat{j}+\hat{k}$,is $1$ cubic unit. If $\theta$ is the angle between $\bar{u}$ and $\bar{w}$,then the value of $\cos \theta$ is:

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{b}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\overrightarrow{c}$ be a vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$. Then $\overrightarrow{a} \cdot ((\overrightarrow{c} \times \overrightarrow{b})-\overrightarrow{b}-\overrightarrow{c})$ is equal to:

If $\overrightarrow{a} = 2\hat{i} + \hat{j} + 3\hat{k}$,$\overrightarrow{b} = 3\hat{i} + 3\hat{j} + \hat{k}$ and $\overrightarrow{c} = c_{1}\hat{i} + c_{2}\hat{j} + c_{3}\hat{k}$ are coplanar vectors and $\overrightarrow{a} \cdot \overrightarrow{c} = 5$,$\overrightarrow{b} \perp \overrightarrow{c}$,then $122(c_{1} + c_{2} + c_{3})$ is equal to.......

For three vectors $a, b, c$,the value of $[a \times b, b \times c, c \times a]$ is equal to:

Let $a, b, c$ be three distinct real numbers,none equal to $1$. If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+ c \hat{k}$ are coplanar,then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to