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.
the Geometric Mean of $$a$$ and $$b$$
B.
the Arithmetic Mean of $$a$$ and $$b$$
C.
equal to zero
D.
the Harmonic Mean of $$a$$ and $$b$$
Answer :
the Geometric Mean of $$a$$ and $$b$$
Solution :
Vector $$a\vec i + a\vec j + c\vec k,\,\vec i + \vec k$$ and $$c\vec i + c\vec j + b\vec k$$ are coplanar
\[\left| \begin{array}{l}
a\,\,\,\,\,\,\,a\,\,\,\,\,c\\
1\,\,\,\,\,\,0\,\,\,\,\,\,1\\
c\,\,\,\,\,\,\,c\,\,\,\,\,\,b
\end{array} \right| = 0\,\,\, \Rightarrow {c^2} = ab\,\,\, \Rightarrow c = \sqrt {ab} \]
$$\therefore \,\,c$$ is G.M. of $$a$$ and $$b.$$