Let $a, b$ and $c$ be three vectors. Then the scalar triple product $[a, b, c]$ is equal to:

If $x$ and $y$ are real numbers such that $\hat{i}+\hat{j}+\hat{k}$, $-2 \hat{i}+3 \hat{j}+2 \hat{k}$, $x \hat{i}-5 \hat{j}+3 \hat{k}$, and $\hat{i}+y \hat{j}-\hat{k}$ are the position vectors of four coplanar points, then the locus of $P(x, y)$ is

Let $x_{0}$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c}),$ where $\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}$,$\vec{b}=-2 \hat{i}+x \hat{j}-\hat{k}$ and $\vec{c}=7 \hat{i}-2 \hat{j}+x \hat{k}$. Then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ at $x=x_{0}$ is:

If three vectors $2\hat{i}-\hat{j}-\hat{k}$,$\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda\hat{j}+5\hat{k}$ are coplanar,then the value of $\lambda$ is

$(\bar{a}+2 \bar{b}-\bar{c}) \cdot \{(\bar{a}-\bar{b}) \times (\bar{a}-\bar{b}-\bar{c})\} = $

If $\overline{a}=\hat{i}-\hat{k}$,$\overline{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$ and $\overline{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$,then $\overline{a} \cdot(\overline{b} \times \overline{c})$ depends on