The value of $m$,if the vectors $\hat{\imath}-\hat{\jmath}-6 \hat{k}$,$\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$,and $2 \hat{\imath}-5 \hat{\jmath}+m \hat{k}$ are coplanar,is

If $\vec{a} = \hat{i} - \hat{k}$, $\vec{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}$ 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 $a, b, c$ are non-negative distinct numbers and $a \hat{\imath}+a \hat{\jmath}+c \hat{k}$,$\hat{\imath}+\hat{k}$ and $c \hat{\imath}+c \hat{\jmath}+b \hat{k}$ are coplanar vectors,then

The volume of the tetrahedron,whose vertices are given by the vectors $-i + j + k$,$i - j + k$,and $i + j - k$ with reference to the fourth vertex as the origin,is

If $\vec{r}$ is a vector perpendicular to both the vectors $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and $3 \hat{i}-\hat{j}+\hat{k}$ and satisfies $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+4 \hat{k})=5$, then $|\vec{r}|=$

For three vectors $u, v, w$,which of the following expressions is not equal to any of the remaining three?