If the volume of a tetrahedron,whose vertices are with position vectors $\hat{i}-6 \hat{j}+10 \hat{k}$,$-\hat{i}-3 \hat{j}+7 \hat{k}$,$5 \hat{i}-\hat{j}+\lambda \hat{k}$ and $7 \hat{i}-4 \hat{j}+7 \hat{k}$ is $11$ cubic units,then the value of $\lambda$ is:

If $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular vectors such that $|\bar{a}| = a, |\bar{b}| = b, |\bar{c}| = c$,then $[\bar{a} \bar{b} \bar{c}] = ......$

If the vectors $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}=x\hat{i}+(x-2)\hat{j}-\hat{k}$ are coplanar, then $x=$

Let $\bar{a}$ and $\bar{c}$ be unit vectors at an angle $\frac{\pi}{3}$ with each other. If $(\bar{a} \times(\bar{b} \times \bar{c})) \cdot(\bar{a} \times \bar{c})=5$,then $\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]=$

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:

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$,$\vec{b}=\hat{i}+3 \hat{j}-\hat{k}$ and $\vec{c}=3 \hat{i}-\hat{j}-2 \hat{k}$,then the value of $\left|\begin{array}{lll}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \\ \vec{c} \cdot \vec{a} & \vec{c} \cdot \vec{b} & \vec{c} \cdot \vec{c}\end{array}\right|$ is: