Let a, b and c be the distinct non-negative numbers. If the vectors `a hat(i)+c hat(k), hat(i)+hat(k), c hat(i) + c hat(j)+bhat(k)` lie on a plane, then which one of the following is correct?

Options

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

Answer

B

Explanation

Given `a hat(i)+a hat(j)+chat(k), hat(i)+hat(k).and c hat(i)+chat(j)+b hat(k)` lie on a plane.
`rArr ` vectors `a hat(i)+a hat(j)+chat(k), hat(i)+hat(k) and c hat(i)+chat(j)+bhat(k)` are coplanar.
`:. |(a,a,c),(1,0,1),(c,c,b)|=0`
`rArr a(-c)-a(b-c)+c(c)=0`
`rArr -ac-ab+ac+c^(2)=0`
`rArr c^(2)=ab`
`rArr c` is the geometric mean of a and b.