let `veca , vecb and vecc` be three vectors having magnitudes 1, 1 and 2, respectively, if `vecaxx(vecaxxvecc)+vecb=vec0, ` then the acute angle between `veca and vecc` is _______

Options

A. `(pi)/4`
B. `(pi)/6`
C. `(pi)/3`
D. none of these

Answer

B

Explanation

We have `|veca|=1, |vecb|=1` and `|vecc|=2`
Also
`vecaxx(vecaxxvecc)+vecb=vec0`
`implies(veca.vecc)veca-(veca.veca)vecc+vecb=vec0`
`implies(veca.vecc)veca-vecc+vecb=vec0 [ :' veca.veca=|veca|^(2)=1]`
`implies(veca.vecc)veca-vecc=-vecb`
`implies|(veca.vecc)veca-vecc|=|-vecb|`
`implies|(veca.vecc)veca-vecc|^(2)=|vecb|^(2)`
`=|(veca.vecc)veca|^(2)+|vecc|^(2)-2{(veca.vecc)veca.vecc}=|vecb|^(2)`
`implies(veca.vecc)^(2)|veca|^(2)+|vecc|^(2)-2(veca.vecc)(veca.vecc)=|vecb|^(2)`
`implies(veca.vecc)^(2){|veca|^(2)-2}+|vecc|^(2)=|vecb|^(2)`
`implies-(veca.vecc)^(2)+4=1`
`implies(veca.vecc)^(2)=3`
`impliesveca.vecc=+-sqrt(3)`
`implies|veca||vecc|cos theta=sqrt(3)`
Where `theta` is an acute angle between `veca` and `vecc`
`impliescos theta=(sqrt(3))/2impliestheta=pi//6`