Question

Let $a,b$ and $c$ 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. Geometric Mean of $a$ and $b$
• B. Arithmetic Mean of $a$ and $b$
• C. Equal to zero
• D. Harmonic Mean of $a$ and $b$

Answer 1

The correct option is A Geometric Mean of $a$ and $b$

Vector $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k},$ are coplanar

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

$\Rightarrow -1(ab-c^2)-1(ac-ac)=0$

$\Rightarrow c^2=ab$

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