`veca and vecb` are two vectors such that `|veca|=1 ,|vecb|=4 and veca.` Vecb `=2 . If `vecc` = (2vecaxx vecb) - 3vecb` then find angle between `vecb and vecc`.

Options

A. `pi/3`
B. `pi/6`
C. `(3pi)/4`
D. `(5pi)/6`

Answer

d

Explanation

`|veca|=1 , |vecb|=4 ,veca.vecb =2`
` vecc= ( 2veca xx vecb) - 3 vecb`
` vecc + 3vecb = 2veca xx vecb `
` veca .vecb =2 `
` |veca|. |vecb| cos theta =2`
`or cos theta 2/ (|veca|.|vecb|) = 2/4`
` = 1/2`
` or = pi/3`
`|vecc|+ 3vecb|^(2)= |2veca xx vecb|^(2)`
` or |vecc|^(2) +9 |vecb|^(2) + 2vecc. 3vecb= 4 |veca|^(2) 3vecb= 4 |veca|^(2) |vecb|^(2) sin^(2) theta`
`|vecc|^(2) + 144 + 6 vecb.vecc= -48`
` or |vec|^(2) + 96 + 6 (vecb.vecc) =0`
`or vecc = 2veca xx vecb - 3vecb`
`Rightarrow vecb. vecc = 0 -3 xx 16`
=-48 ,
Putting value of b `vecb . vecc `in Eq. (i) we have
`|vecc|^(2) + 96 - 6 xx 48`
` = 0 = 48xx 4 = 192`
Again , putting the value of `|vecc|` in Eq. (i) we have
`192 + 96 + 6|vecb|.|vecc| cos alpha=0`
` or cos alpha = 288 /( 6 xx 4 xx 8 sqrt3) = - 3/ (2sqrt3)`
` = sqrt3/2`
`or alpha = (5pi)/6`