Two vectors `vecA` and `vecB` are defined as `vecA=ahati` and `vecB=a( cos omegahati+sin omega hatj)`, were a is a constant and `omega=pi//6 rads^(-1)`. If `|vecA+vecB|=sqrt(3)|vecA-vecB|` at time `t=tau` for the first time, the value of `tau`, in seconds , is _________

Answer

`[2.00]`

Explanation

[2.00]: `vecA=ahati and vecB=a cos omega thati+a sin omega thatj`
`|vecA+vecB|=sqrt3|vecA-vecB|`
`rArr sqrt((a+ a cos omegat)^(2)+(a sin omega t)^(2))`
`=sqrt3 sqrt((a-acos omegat)^(2)+(a sin omega t)^(2))`
`rArr a^(2)+a^(2) cos ^(2) omegat+a^(2) sin^(2) omegat`
`=3(a^(2)+a^(2) cos ^(2) omega t-2a^(2) cos omega t+a^(2) sin ^(2) omegat)`
`rArr 2a^(2) +2a^(2) cos omega t =3 (2a^(2)-2a^(2) cos omegat)`
`rArr a^(2)+a^(2) cos omega t =3a^(2)-3a^(2) cos omega t`
`rArr 4a^(2) cos omega t =2a^(2)`
`rArr cos omega t=(1)/(2)`
`rArr omega t =pi//3`
`rArr (pi)/(6) t=(pi)/(3) rArr t=2s`