Question
Let $$a,\,b,\,c$$ be three distinct positive real numbers. If $$\overrightarrow p ,\,\overrightarrow q ,\,\overrightarrow r $$ lie in a plane, where $$\overrightarrow p = a\overrightarrow i - a\overrightarrow j + b\overrightarrow k ,\,\overrightarrow q = \overrightarrow i + \overrightarrow k $$ and $$\overrightarrow r = c\overrightarrow i + c\overrightarrow j + b\overrightarrow k ,$$ then $$b$$ is :
A.
the AM of $$a,\,c$$
B.
the GM of $$a,\,c$$
C.
the HM of $$a,\,c$$
D.
equal to $$0$$
Answer :
the HM of $$a,\,c$$
Solution :
\[{\rm{Here }}\left[ {\overrightarrow p \,\,\overrightarrow q \,\,\overrightarrow r } \right] = 0.{\rm{ \,So, }}\left| \begin{array}{l}
a\,\,\,\,\, - a\,\,\,\,\,\,b\\
1\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,1\\
c\,\,\,\,\,\,\,\,\,\,\,c\,\,\,\,\,\,\,\,\,\,b
\end{array} \right| = 0\]
$$\eqalign{
& {\text{or }}a\left( {0 - c} \right) + a\left( {b - c} \right) + b\left( {c - 0} \right) = 0 \cr
& \Rightarrow \,ab + bc = 2ac \cr
& \therefore \,b = \frac{{2ac}}{{a + c}} \cr} $$
So, $$b$$ is the HM of $$a,\,c.$$