If the points whose position vectors are $3i - 2j - k,$ $2i + 3j - 4k,$ $-i + j + 2k,$ and $4i + 5j + \lambda k$ lie on a plane,then $\lambda = $

The volume of the tetrahedron having vertices $(1, -6, 10)$,$(-1, -3, 7)$,$(5, -1, \lambda)$ and $(7, -4, 7)$ is $11 \text{ cubic units}$. Then $\lambda = $

$(a+b) \cdot(b+c) \times(a+b+c)$ is equal to

If the 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

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

If $A \equiv (1, -1, 0)$,$B \equiv (0, 1, -1)$,and $C \equiv (-1, 0, 1)$,then the unit vector $\overline{d}$ such that $\overline{a}$ and $\overline{d}$ are perpendicular and $\overline{b}, \overline{c}, \overline{d}$ are coplanar is