if `overset(to)(b) " and " overset(to)(c )` are any two non- collinear unit vectors and `overset(to)(a)` is any vector then
`(overset(to)(a).overset(to)(b))overset(to)(b).(overset(to)(a).overset(to)(c )) overset(to)(c ) + .(overset(to)(a).(overset(to)(b)xxoverset(to)(c)))/(|overset(to)(b)xxoverset(to)(c)|^(2)).(overset(to)(b)xxoverset(to)(c))=.........`

Answer

`overset(to)(a)`

Explanation

Let `hat(i) ` be a unit vector in the direction of `vec(b) , hat(j)` in the direction of `vec(c )` . Note that `vec( c) = hat(j)`
and `(vec(b) xx vec(c )) = |vec(b)||vec(c )| " sin " alpha hat(k) = " sin " alpha hat(k)`
where `hat(k) ` is a vector perpendicular to `vec(b) " and " vec(c )`
`rArr |vec(b) xx vec(c )| " = sin " alpha rArr hat(k) = (vec(b) xx vec( c))/(|vec(b) xx vec(c )|)`
Let `vec( a) = a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k)`
Now `vec(a) ". " vec(b) = vec(a) ". " hat(i) = hat(i) "." (a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k)) = a_(1)`
and `vec(a) ". " vec(c ) = vec(a) " ." hat(j) = hat(j) ". " (a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k)) = a_(2)`
and `vec(a) ". " (vec(b) xx vec( c))/( |vec(b) xx vec( c)|) = vec(a) ". "vec(k) = a_(3) `
`:. (vec(a) ". " vec(b)) vec(b) + (vec(a) ". " vec(c )) vec( c) + (vec(a) (vec(b)xx vec(c )))/(|vec(b) xx vec(c )|^(2)) (vec(b) xx vec(c ))`
`=a_(1) vec(b) + a_(2) vec(c ) + a_(3) .((vec(b) xx vec(c )))/(|vec(b) xx vec(c )|) = a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) = hat(a)`