Question

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

• A. Equal to zero
• B. The harmonic mean of $a$ and $b$
• C. The geometric mean of $a$ and $b$
• D. The arithmetic mean of $a$ and $b$

Answer 1

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

Since the given vectors are coplanar

$\therefore \left|\begin{array}{ccc}a& a& c\\ 1& 0& 1\\ c& c& b\end{array}\right|=0$

Applying $C_2\to C_2-C_1$

$\Rightarrow \left|\begin{array}{ccc}a& 0& c\\ 1& -1& 1\\ c& 0& b\end{array}\right|=0$

Expanding along $C_2$, we get.

$\Rightarrow ab-c^2=0\Rightarrow ab=c^2$

$\therefore a,b,c$ are in $G.P.$