Question

Let a, b, c be distinct non-negative numbers. If the vectors a$\hat i$ + a$\hat j$ + c$\hat k$, $\hat i$ + $\hat k$ and c$\hat i$ + c$\hat j$ + b$\hat k$ lie in a plane, then c is

• A. not arithmetic mean of a and b.
• B. the geometric mean of a and b.
• C. the arithmetic mean of a and b
• D. the harmonic mean of a and b.

Answer 1

Answer: (B)

the geometric mean of a and b.

Answer 2

Let a = a$\hat i$ + a$\hat j$ + c, b = + $\hat k$ and c = c$\hat i$ + c$\hat j$ + b$\hat k$
a,b and c lies in a plane, if [a b c]=0,
$\begin{vmatrix} a & a & c\\\ 1 & 0 & 1 \\\ c & c & b\end{vmatrix}$ =0
now apply C1 $\to$ C1 - C2
$\begin{vmatrix} 0 & a & c\\\ 1 & 0 & 1 \\\ 0 & c & b\end{vmatrix}$ = 0
1⋅[ab−c2] = 0
ab = c2 which means c is geometric mean of a and b.