Question

Let `overset(to)(a) , overset(to)(b) " and " overset(to)(c )` be three vectors having magnitudes 1 , and 2 respectively . If `overset(to)(a) xx (overet(to)(a) xx overset(to)(c ) ) + overset(to)(b) = overset(to)(0)` then the actue angle between `overset(to)(a) " and " overset(to)(c )` is ......

Answer 1

Correct Answer - `(pi)/(6)`
Given `vec(a) xx (vec(a) xx vec(c )) +vec(b) =vec(0)`
`rArr (vec(a) ". " vec(c )) vec(a) - (vec(a) "." vec(a)) vec( c) + vec(b) =vec(0)`
`rArr (2 cos 0) vec(a) -vec(c ) + vec(b) =vec(0)`
`rArr (2 cos 0 vec(a) - vec( c))^(2) = (-vec(b))^(2)`
`rArr 4 cos^(2) 0 . |vec(a)|^(2)+|vec( c)|^(2) 2 2 cos 0 vec(a) ". " vec(c ) = |vec(b)|^(2)`
`rArr 4 cos^(2) 0-4 -8 cos^(2) 0=1`
` rArr 4 cos^(2) 0=3`
`rArr cos 0 = +- (sqrt(3))/(2)`
`rArr cos 0 = (sqrt(3))/(2) rArr 0= (pi)/(6)`
For 0 to be acute