If `overset(to)(a),overset(to)(b),overset(to)(c ),overset(to)(d)` are four distinct vectors satisfying the conditions `overset(to)(a)xxoverset(to)(b)=overset(to)(c )xx overset(to)(d) " and " overset(to)(a)xxoverset(to)(c ) = overset(to)(b)xx overset(to)(d)` then prove that `, overset(to)(a).overset(to)(b)+overset(to)(c ). overset(to)(d) ne overset(to)(a). overset(to)(c)+overset(to)(b).overset(to)(d)`.

Explanation

Given `vec(a) xx vec(b) xx vec(d) " and " vec(a) xx vec( c) = vec(b) xx vec(d)`
`rArr vec(a) xx vec(b) - vec(a) xx vec( c) = vec( c) xx vec(d) - vec(b) xx vec(d)`
`rArr vec(a) xx (vec(b) - vec( c)) = (vec(c )- vec(b)) xx vec(d)`
`rArr vec(a) xx (vec(b) -vec(c )) -(vec(c )-vec(b)) xx vec(d) =0`
`rArr vec(a) xx (vec(b) -vec(c )) - vec(d) xx (vec(b)-vec(c )) =0`
`rArr (vec(a) - vec(d)) xx (vec(b)-vec(c )) =0 rArr (vec(a) -vec(d))||(vec(b)-vec(c ))`
`:. (vec(a) xx vec(d)) .(vec(b) - vec(c )) ne 0`
`rArr vec(a) "." vec(b) = vec(d) ". " vec(c ) ne vec(d) ". " vec(b) + vec(a) ". " vec(c )`