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 $c\hat{i}+c\hat{j}+b\hat{k}$, are coplanar then c is

• A. AM of a and b
• B. GM of a and b
• C. HM of a and b
• D. None of the above

Answer 1

The correct option is B GM of a and b

Since, the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are coplanar, therefore
$\left|\begin{array}{ccc}a& a& c\\ 1& 0& 1\\ c& c& b\end{array}\right|=0$
$\Rightarrow -ac-ab+ac+c^2=0$
$\Rightarrow c^2=ab$
$\Rightarrow$ c is the GM of a and b.