Let a, b and c be distinct non-negative numbers. If the vectors `ahati+ahatj+c hatk, hati+hatk and chati+chatj+bhatk` lie in a plane, then c is

Options

A. the harmonic mean of a and b
B. equal to zero
C. the arithmetic mean of a and b
D. the geometric mean of a and b

Answer

D

Explanation

Since , the given points lie on a plane.
`:. |{:(a,a,c),(1,0,1),(c,c,b):}| = 0`
On applying `C_(1) rarr C_(1) - C_(2)` , we get
`|{:( 0,a,c),(1,0,1),(0,c,b):}| = 0`
`rArr -1(ab - c^(2)) = 0 rArr c^(2) = ab`
Hence, c is GM of a and b.