Question

Let $a,b,c$ be distnict non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+b\hat{k},\hat{i}+\hat{k}$ and $b\hat{i}+b\hat{j}+c\hat{k}$ lie in a plane ,then the correct option among the following is

• A. $a,b,c$ are in $A.P.$
• B. $a,b,c$ are in $G.P.$
• C. $a,c,b$ are in $H.P.$
• D. $a,c,b$ are in $G.P.$

Answer 1

The correct option is B $a,b,c$ are in $G.P.$

Since vectors are coplanar, scalar tripple product is zero.
$\Rightarrow \left|\begin{array}{ccc}a& a& b\\ 1& 0& 1\\ b& b& c\end{array}\right|=0\Rightarrow -ab-a(c-b)+b^2=0\Rightarrow b^2=ac$
So, $a,b,c$ are in $G.P.$