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 $$c\hat i + c\hat j + b\hat k$$ lie in a plane, then $$c$$ is :
A.
the Arithmetic Mean of $$a$$ and $$b$$
B.
the Geometric Mean of $$a$$ and $$b$$
C.
the harmonic Mean of $$a$$ and $$b$$
D.
equal to zero
Answer :
the Geometric Mean of $$a$$ and $$b$$
Solution :
$$a,\,b,\,c$$ are distinct non negative numbers and 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.
\[\therefore \left| \begin{array}{l}
a\,\,\,\,a\,\,\,\,c\\
1\,\,\,\,0\,\,\,\,1\\
c\,\,\,\,c\,\,\,\,b
\end{array} \right| = 0 \Rightarrow \left| \begin{array}{l}
a\,\,\,\,a\,\,\,\,c - a\\
1\,\,\,\,0\,\,\,\,\,\,\,0\\
c\,\,\,\,c\,\,\,\,b - c
\end{array} \right|\]
Operating $${C_3} \to {C_3} - {C_1}$$
Expanding along $${R_2},$$ we get
\[ - \left| \begin{array}{l}
a\,\,\,\,c - a\\
c\,\,\,\,b - c
\end{array} \right| = c\left( {c - a} \right) - a\left( {b - c} \right) = 0\]
$$\eqalign{
& \Rightarrow {c^2} - ac - ab + ac = 0 \cr
& \Rightarrow {c^2} = ab \Rightarrow a,\,c,\,b\,\,{\text{are in G}}{\text{.P}}{\text{.}} \cr} $$
$$\therefore \,\,c$$ is the Geometric Mean of $$a$$ and $$b.$$