Let a,b,c be distinct non-negative numbers.

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:

Accepted Answer

$$\begin{vmatrix}a&a&c\1&0&1\c&c&b\end{vmatrix}=0 \implies -ab+c^2=0 \implies c^2=ab \implies c=\sqrt{ab}$$

So $c$ is the Geometric Mean of $a$ and $b$.

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