Question

Let $a,b$ and $c$ be distinct non-negative numbers. If 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. Arithmetic mean of $a$ and $b$
• B. Geometric mean of $a$ and $b$
• C. Harmonic mean of $a$ and $b$
• D. Equal to zero

Answer 1

Answer: (B)

Geometric mean of $a$ and $b$

$a,b$ and $c$ distinct negative numbers and 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

$\left|\begin{array}{ccc}a& a& c\\ 1& 0& 1\\ c& c& b\end{array}\right|=0$
or $ac+c^2-ab-ac=0$
or $c^2=ab$

Hence, $a,b,c$ are in $G.P$.
So $c$ is the $G.M$. of $a$ and $b.$