Let a three dimensional vector ` vec V` satisfy the condition, `2 vec V+ vec Vxx( hat i+2 hat j)=2 hat i+ hat kdot` If `3| vec V|=sqrt(m)dot` Then find the value of `mdot`

Answer

6

Explanation

`2vecV+vecVxx(hati+2hatj) = (2hati+hatk)`
`or 2vecV. (hati+2hatj) +0=(2hati+hatk). (hati+2hatj)`
`or 2vecVr. (hati+2hatj)=2`
`or |vecV. (hati+2hatj)^(2)|=1`
`or |vecV|^(2).|hati +2hatj|^(2) cos^(2)theta=1`
(`theta` is the angle between `vecV and hati+2hatj)`
`or |vecV|^(2)5(1-sin^(2)theta)=1`
`or |vecV|^(2) 5 sin^(2)theta =5|vecV|^(2)-1`
from Eq. (i), we have
`|2vecV+vecVxx(hati+2hatj)|^(2)=|2hati+hatk|^(2)`
`or 4|vecV|^(2)+|vecVxx(hati+2hatj)|^(2)=5`
`or 4|vecV|^(2)+|vecV|^(2).|hati + 2hatj|^(2) sin^(2) theta=5`
`or 4|vecV|^(2)+5|vecV|^(2)sin^(2)theta=5`
`or 4|vecV|^(2)+5|vecV|Y^(2)-1=5`
` 9|vecV|^(2)=6`
`or 3|vecV|=sqrt6`
` = sqrt6 = sqrtm`
m=6