At what angle should the two forces `(vecA+vecB) and (vecA-vecB)` act so that their resultant will be `sqrt(3A^2+B^2)`.

Explanation

If `veca =(vecA+vecB)` , then
`a^2=veca*veca=(vecA+vecB)*(vecA+vecB)=A^2+B^2+2vecA*vecB`
if `vecB=(vecA-vecB)` , similarly
`b^2=vecb*vecb=A^2+B^2-2vecA*vecB`
`veca*vecb=(vecA+vecB)*(vecA-vecB)`
`=A^2-vecA*vecB+vecA+vecB-B^2=A^2-B^2`
If the angle between `veca and vecb` is `alpha ,veca*vecb =ab cos alpha`
`therefore ab cos alpha =A^2-B^2`
if `vecc` be the resultant of `veca and vecb`,
`c^2=a^2+b^2+2ab cos alpha`
`or, 3A^2+B^2=(A^2+B^2+2vecA*vecB)`
`+(A^2+B^2-2vecA*vecB)+2(A^2-B^2)`
`or, 3A^2+B^2=4A^2 or, A^2-B^2 =0`
Hence, `ab cos alpha =A^2-B^2=0`
`or, cos alpha =0 or, alpha =90^@`
`therefore` If the force act at an angle `90^@` their resultant will be `sqrt(3A^2+B^2)`.