I draw a vector \(\vec{r_{1}}\) in the plane, and a vector \(\vec{r_{2}}\) orthogonal to \(\vec{r_{1}}\). Their resultant vector is then \(\vec{R_{1}}\).

Then, I draw a vector \(\vec{r_{3}}\) orthogonal to \(\vec{R_{1}}\) with their resultant being \(\vec{R_{2}}\).

Following this pattern, I keep on drawing vectors and resultants ad infinitum.

If \(\frac{|\vec{r_{n}}|}{|\vec{r_{1}}|} = \frac{1}{n}\) for all \(n \geq 1\) and \(|\vec{R_{\infty}}| = \pi\), find the dot product of \(\vec{r_{1}}\) with itself.

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