If ` hat a , hat b ,a n d hat c` are three unit vectors, such that ` hat a+ hat b+ hat c` is also a unit vector and `theta_1,theta_2` and `theta_3` are angles between the vectors ` hat a , hat b ; hat b , hat ca n d hat c , hat a` respectively, then among `theta_1,theta_2` and `theta_3`. a. all are acute angles b. all are right angles c. at least one is obtuse angle d. none of these

Options

A. all are acute angles
B. all are right angles
C. at least one is obtuse angle
D. none of these

Answer

c

Explanation

`|veca+vecb+vecc|^(2)=1`
`Rightarrow |veca|^(2)+|vecb|^(2) +|vecc|^(2)+2|veca||vecb|costheta_(1)`
` + 2|vecb||vecc|costheta_(2)+2|vecc||veca|costheta_(3)=1`
`Rightarrow costheta_(1)+costheta_(2)+costheta_(3)=-1`
Hence, one of `theta_(1),theta_(2) and theta_(3)` should be an obtuse angle.