Question

If vectors `overset(to)(a) , overset(to)(b) , overset(to)( C)` are coplanar then show that
`|{:(overset(to)(a),,overset(to)(b),,overset(to)(c )),(overset(to)(a)"."overset(to)(a),,overset(to)(a)"."overset(to)(b),,overset(to)(a)"."overset(to)(c )),(overset(to)(b)"."overset(to)(a),,overset(to)(b)"."overset(to)(b),,overset(to)(b)"." overset(to)(c )):}|`

Answer 1

Answer - Given that `vec(a) , vec(b) vec(c )` are coplanar vectors
`:. ` There exists scalar x,y, z not all zero such that
`x vec(a ) + yvec(b) + z vec(c )=0`
Taking dot with `vec(a) " and " vec(b)` respectively we get
`x (vec(a) " ." vec(b)) + y(vec(a) " ." vec(b)) + z (vec(a) ". " vec( c))=0`
and `x (vec(a) ". " vec(b )) + y(vec( b) "." vec(b )) + z (vec( c) ". " vec( b))=0`
Since Eqs . (i) (ii) and (iii) represent homogeneous
equations with `(x,y,z) ne (0,0,0)`
`rArr ` Non-trivial solutions
`:. Delta =0 rArr |{:(vec(a),,vec(b),,vec( c)),(vec(a) ". " vec(a),,vec(a) ". "vec(b),,vec(a)"."vec(c)),(vec(b) "."vec(b),,vec(b) "."vec(b),,vec(b)"."vec(c )):}|=vec(0)`