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 _______

Answer

`pi//6`

Explanation

`veca xx (vecaxxvecc) +vecb=vec0`
`or (veca.vecc)veca-(veca.veca)vecc+vecb=vec0`
`or 2 cos theta. veca-vecc+vecb=vec0`
(using `|veca|=1,|vecb|=1,|vecc|=2`)
`or (2costheta veca-vecc)^(2)=(-vecb)^(2)`
`or 4cos^(2)theta.|veca|^(2)+|vecc|^(2)`
`-2.2 cos theta.veca.vecc=|vecb|^(2)`
`or 4cos^(2)theta+4-8costheta.costheta=1`
`or cos^(2)theta-8cos^(2)theta+4=1`
`or 4 cos^(2) theta=3`
`or cos theta = +- sqrt3//2`
for `theta` to be acute , `cos theta= sqrt3/2 Rightarrow theta= pi/6`