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If a,b and c are three unit vectors equa...

If a,b and c are three unit vectors equally inclined to each other at angle `theta`. Then, angle between a and the plane of b and c is

A

`cos^(-1)((costheta)/(cos(theta//2)))`

B

`sin^(-1)((sintheta)/(sin(theta//2)))`

C

`sin^(-1)((costheta)/(cos(theta//2)))`

D

`cos^(-1)((sintheta)/(sin(theta//2)))`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `alpha` be angle between a and angular bisector of b and c. then
`cosalpha=(a*(b+c))/(|a||b+c|)`
`=(|a||b|costheta+|a||vec(c)|costheta)/(sqrt(|vec(b)|^(2)+|c|^(2)+2|b||c|)costheta)`
`=(2costheta)/(sqrt(2+2costheta))=(costheta)/(costheta//2)" "[|vec(a)|=|vec(b)|=|vec(c)|=1]`
`rArr" "alpha=cos^(-1)((costheta)/(cos((theta)/(2))))`
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