`bara` & `barb` be two vectors such that `|a|=1,|b|=4`& `bara.barb=2`. If `barc=(2baraxxbarb)-3barb`, then which of the following is/are correct?

Options

A. `barb.barc=48`
B. `barb.barc=-48`
C. Angle between `barb` & `barc` is`(5pi)/6`
D. Angle between `barb` & `barc` is `(pi)/6`.

Answer

B, C

Explanation

`|bara|=1, |barb|=4, bara.barb=2` & `barc+3barb=2baraxxbarb`
`Q bara.barb=2`
`impliescostheta=2/(|bara||barb|)=1/2 impliestheta =(pi)/3`
Agai `|barc+3barb|^(2)=|2baraxxbarb|^(2)`
`implies |barc|^(2)+9|barb|^(2)+2barc.3barb=4|bara|^(2)|barb|^(2)sin^(2)theta`
`implies|barc|^(2)+144+6barb.barc=48`
Also, `barc=2baraxxbarb-3barb`
`impliesbarc.barc=barb(2baraxxbarb)-3|barb|^(2)`
`impliesbarc.barc=0-3xx16=-48`
`:.|barc|^(2)+96-6xx48=0`
`=|barc|^(2)=192`
`implies|barc|=8sqrt(3)`
`barb.barc=|barb||barc|=cos alphaimplies -48=4xx8sqrt(3) cos alpha`
`impliescos alpha=-6/(4sqrt(3))=-(sqrt(3))/2`
`impliesalpha=(5pi)/6`