`vec(V) = (vec(a) xx vec(b)) xx (vec(c) xx vec(d)) `

`= vec(u) xx (vec(c) xx vec(d)) = [ vec(a)vec(b)vec(d)] vec(c) - [vec(a)vec(b)vec(c)] vec(d) ` .................`(1)` (where `vec(u) = vec(a) xx vec(b)`)

again `vec(V) = (vec(a) xx vec(b)) xx underbrace (vec(c) xx vec(d))_(vec(v)) = (vec(a)*vec(v)) vec(b) - (vec(b)*vec(v)) vec(a) = [ vec(a)vec(c)vec(d)] vec(b) - [vec(b)vec(c)vec(d)] vec(a) ` ....................`(2)`

from `(1)` and `(2)`
`[vec(a)vec(b)vec(d)] vec(c) - [vec(a)vec(b)vec(c)] vec(d) = [vec(a)vec(c)vec(d)] vec(b) - [vec(b)vec(c)vec(d)] vec(a)` .......................`(3)`

`text(Note that)`

`(vec(a) xx vec(b)) xx (vec(c) xx vec(d)) =0 =>` planes containing the vectors `vec(a)` &`vec(b)` and `vec(c)` & `vec(d)` are parallel.


`|||^(ly) ` ` (vec(a) xx vec(b)) * (vec(c) xx vec(d)) =0 => ` the two planes a re perpendicular.

`(i)` Equation `(3)` is suggestive that if `vec(a),vec(b),vec(c),vec(d)` are four vectors no `3` three of them are coplanar then each one
of them can be expressed as a linear combination of other.

`(ii)` If `vec(a),vec(b),vec(c),vec(d)` are p.v.'s of four points then these four points are in the same plane if

`[vec(a)vec(b)vec(d)] - [vec(a)vec(b)vec(c)] = [vec(a)vec(c)vec(d)] - [vec(b)vec(c)vec(d)]`