If the vectors $m \hat{i} + m \hat{j} + n \hat{k}$,$\hat{i} + \hat{k}$,and $n \hat{i} + n \hat{j} + p \hat{k}$ lie in a plane,then...

Let $\vec{a} = \hat{i} - \hat{k}$,$\vec{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}$,and $\vec{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k}$. Then the scalar triple product $[\vec{a} \, \vec{b} \, \vec{c}]$ depends on:

If $\overline{u}, \overline{v}$ and $\overline{w}$ are three non-coplanar vectors,then $(\bar{u}+\bar{v}-\bar{w}) \cdot [(\bar{u}-\bar{v}) \times (\bar{v}-\bar{w})]$ is equal to

Consider the vectors $u = a \hat{i} + b \hat{j} + c \hat{k}$,$v = a^2 \hat{i} + b^2 \hat{j} + c^2 \hat{k}$ and $w = a^3 \hat{i} + b^3 \hat{j} + c^3 \hat{k}$. These vectors are coplanar if and only if

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 $3 \hat{i}-2 \hat{j}-\hat{k}$,$2 \hat{i}+3 \hat{j}-4 \hat{k}$,$-\hat{i}+\hat{j}+2 \hat{k}$ and $4 \hat{i}+5 \hat{j}+\lambda \hat{k}$ are respectively the position vectors of four coplanar points $P, Q, R$ and $S$,then $\lambda=$