Let $a, b$ and $c$ be distinct non-negative numbers. If the vectors $a \hat{i}+a \hat{j}+c \hat{k},…
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
the Geometric Mean of $a$ and $b$
the Arithmetic Mean of $a$ and $b$
equal to zero
the Harmonic Mean of $a$ and $b$
Solution
Vector $a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $c \hat{i}+c \hat{j}+b \hat{k}$ are coplanar
$
\begin{aligned}
& \left|\begin{array}{lll}
a & a & c \\
1 & 0 & 1 \\
c & c & b
\end{array}\right|=0 \Rightarrow c^2=a b \\
& \therefore a, b, c \text { are in G.P. }
\end{aligned}
$