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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=?

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=?

Grade:12

1 Answers

Yash Chourasiya
askIITians Faculty 256 Points
10 months ago
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 =abcosθ

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.

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