Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be distinct non-negative umbers. If the vectors $a \hat{i}+a…
Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be distinct non-negative umbers. 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 arithmetic mean of a and b.
the harmonic mean of a and $b$.
the geometric mean of $a$ and $b$.
not arithmetic mean of a and b.
Solution
$\begin{aligned} & \left|\begin{array}{lll}a & a & c \\ 1 & 0 & 1 \\ c & c & b\end{array}\right|=0 \\ & \Rightarrow a b-c^2=0 \\ & \Rightarrow c^2=a b\end{aligned}$
$\Rightarrow$ The geometric mean of a and b is $\mathrm{c}$