If the angles between the vectors `veca` and `vecb,vecb` and `vecc,vecc` an `veca` are respectively `pi/6,pi/4` and `pi/3`, then the angle the vector `veca` makes with the plane containing `vecb`and `vecc`, is

Options

A. `cos^(-1)sqrt(1-sqrt(2//3))`
B. `cos^(-1)sqrt(2-sqrt(3//2))`
C. `cos^(-1)sqrt(sqrt(3//2)-1)`
D. `cos^(-1)sqrt(sqrt(2//3))`

Answer

B

Explanation

Let `theta` be angle between `veca` and plane containing `vecb` and `vecc`.
`therefore 90^(@)-theta`= angle b/w `veca` and `vecb xx vecc`
`|hata xx (hatb xx hatc)|^(2)=|(hata.hatc)hatb-(hata.hatb)hatc|^(2)`
`=sin^(2)(90^(@)-theta)sin^(2)pi/4 = |(cospi/3)hatb-(cospi/6)hatc|^(2)`
`rArr cos^(2)theta xx 1/2 =1/4 +3/4-2 xx 1/2 xx sqrt(3)/2 xx 1/sqrt(2)`
`=1-sqrt(3)/2sqrt(2)`
`rArr cos^(2)theta=2-sqrt(3//2)`