If position vectors a,b,c,d are coplanar, (sinx)a +(2sin2y)b +(3sin3z)c-d =0, min value of sin square x + sin square 2y + sin square 3z=?

Hello Student

Given that vector a,b,c,d are coplanar
So, sum of coefficient=0
sinx + 2sin2y + 3sin3z = 1......................(1)

as from the pattern LHS can also be called to be dot product of two vectors(1,2,3)and(sinx, sin2y ,sin3z)
dot product of these two vectors is
sinx + 2sin2y + 3sin3z

a.b =∣a∣∣b∣cosθ

sinx + 2sin2y + 3sin3z = (sin²x + sin²2y + sin²3z) (14)cosθ...............(2)

From eq. (1) and (2), we get

1/14cosθ =sin²x + sin²2y + sin²3z

So minimum value will be 1/14.

I hope this answer will help you.