# If $a,b,c$ are non negative and distinct numbers and 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 ?

$\begin{array}{1 1} the \;A.M. \;of\; a\:\:and \:\:b \\ the\; G.M. \; of\; a\:\:and \:\:b \\ the\; H.M. \; of \;a\:\:and \:\:b \\ 0 \end{array}$

Given: The three vectors are coplanar.
$\therefore\:\left |\begin {array}{ccc}a & a & c\\1 &0 & 1\\c & c& b\end {array}\right|=0$
$\Rightarrow\:-ac-a(b-c)+c^2=0$
$\Rightarrow\:ab=c^2$
$\Rightarrow\:c$ is G.M. of $a\:and\:b$