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

The Correct Option is B

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.