# let the unit vectors a and b be perpendicular to each other and unit vector c be inclined at an angle theta to both a and b if c=xa+yb+z(a×b) then (1)x=costheta,y=sintheta,z=cos2theta (2)x=sintheta,y=costheta,z=-cos2theta (3)x=y= costheta,z^2=cos2theta (4)x=y=costheta,z^2=-cos2theta

Arun
25758 Points
4 years ago
Dear Akshat

we have c =xa + yb +z (a x b)
c.a = x
and
c.b = y
hence x = y = cos theta

now c.c = c^2
(xa + yb +z (a x b)).(xa + yb +z (a x b)) =  |c|^2

2 x^2 +z^2 |a|^2 |b|^2 – (a.b)^2 = 1
2 x^2 + z^2 – 0 = 1
since a and b are perpendicular to each other

hence z^2  = 1 – 2x^2 = 1- 2 cos^2theta = – cos 2$\theta$

hence option 4 is correct

regards