If the vectors $\overrightarrow{a}+\lambda \overrightarrow{b}+3 \overrightarrow{c}$,$-2 \overrightarrow{a}+3 \overrightarrow{b}-4 \overrightarrow{c}$ and $\overrightarrow{a}-3 \overrightarrow{b}+5 \overrightarrow{c}$ are coplanar,then the value of $\lambda$ is

If the vectors $p \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+q \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r \hat{k}$ $(p \neq q \neq r \neq 1)$ are coplanar,then the value of $pqr-(p+q+r)$ is

If the points with position vectors $\hat{i}-2 \hat{j}+3 \hat{k}$, $2 \hat{i}+3 \hat{j}-4 \hat{k}$, $-3 \hat{i}+\hat{j}-5 \hat{k}$, and $a \hat{i}-2 \hat{j}+4 \hat{k}$ are coplanar, then $a=$

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 $|\lambda \hat{i}-3 \lambda \hat{j}+3 \hat{k}|=$

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:

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