Calculus Level 5

I draw a vector r1\vec{r_{1}} in the plane, and a vector r2\vec{r_{2}} orthogonal to r1\vec{r_{1}}. Their resultant vector is then R1\vec{R_{1}}.

Then, I draw a vector r3\vec{r_{3}} orthogonal to R1\vec{R_{1}} with their resultant being R2\vec{R_{2}}.

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

If rnr1=1n\frac{|\vec{r_{n}}|}{|\vec{r_{1}}|} = \frac{1}{n} for all n1n \geq 1 and R=π|\vec{R_{\infty}}| = \pi, find the dot product of r1\vec{r_{1}} with itself.


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