Let `veca=2hati+hatj-hatk&vecb=hati+2hatj+hatk` be two vectors. Consider a vector `vecC=alphaveca+betavecb,alpha,beta in R`. If the projection of `vec c` on the vector `(veca+vecb)` is `3sqrt(2)` then the minimum value of `(vec c -(vecaxxvecb)).vec c` equal to

Answer

A

Explanation

`vec c=alpha(2hati+hatj-hatk)+beta(hati+2hatj+hatk)`
`impliesvec c=(2alpha+betahati+(alpha=2beta)hatj+(beta-alpha)hatk`
`(vecc.(veca+vecb))/(|veca+vecb|)=3sqrt(2)implies9(alpha+beta)=18impliesalpha+beta=2` ltBrgt `(vecc-vecaxxvecb).vecc=(alphaveca+betavecb-vecaxxvecb).(alphaveca+betavecb)=6alpha^(2)+6alphabeta+6beta^(2)=6[alpha^(2)+alpha(2-alpha)+(2-alpha)^(2)]`
`=6(alpha^(2)-2alpha+4)=min` value 18.