Question

Let $a,b$ and $c$ be distinct non-negative numbers. If the vectors $ai+aj+ck,i+k$ and $ci+cj+bk$ lie in a plane, then $c$ is

• 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 1

The correct option is D The geometric mean of $a$ and $b$

Let $a=ai+aj+ck,b=i+k$ and $c=ci+cj+bk$ $a,b$ and $c$ lies in a plane, if $\left[\begin{array}{ccc}a& b& c\end{array}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}a& a& c\\ 1& 0& 1\\ c& c& b\end{array}\right|=0$
Applying $C_1\to C_1-C_2$
$\Rightarrow \left|\begin{array}{ccc}0& a& c\\ 1& 0& 1\\ 0& c& b\end{array}\right|=0$
$\Rightarrow 1\cdot [ab-c^2]=0$
$\Rightarrow ab=c^2$
Therefore, $c$ is geometric mean of $a$ and $b$.