# Magnitude

Calculus Level 5

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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