Find the equation of plane passing through $$(1, 1, 1)$$ & parallel to the lines $${L_1},{L_2}$$ having direction ratios $$(1,0,-1),(1,-1,0).$$ Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.

A parallelopiped $$'S'$$ has base points $$A, B, C$$ and $$D$$ and upper face points $$A',$$ $$B',$$ $$C'$$ and $$D'.$$ This parallelopiped is compressed by upper face $$A'B'C'D'$$ to form a new parallelopiped $$'T'$$ having upper face points $$A'',B'',C''$$ and $$D''.$$ Volume of parallelopiped $$T$$ is $$90$$ percent of the volume of parallelopiped $$S.$$ Prove that the locus of $$'A''',$$ is a plane.

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are distinct vectors such that
$$\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d \,.$$ Prove that
$$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) \ne 0\,\,i.e.\,\,\,\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c \ne \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $$

$${P_1}$$ and $${P_2}$$ are planes passing through origin. $${L_1}$$ and $${L_2}$$ are two line on $${P_1}$$ and $${P_2}$$ respectively such that their intersection is origin. Show that there exists points $$A, B, C,$$ whose permutation $$A',B',C'$$ can be chosen such that (i) $$A$$ is on $${L_1},$$ $$B$$ on $${P_1}$$ but not on $${L_1}$$ and $$C$$ not on $${P_1}$$ (ii) $$A'$$ is on $${L_2},$$ $$B'$$ on $${P_2}$$ but not on $${L_2}$$ and $$C'$$ not on $${P_2}$$