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.

The answer is 6.

1 solution

Jake Lai
Jan 30, 2015

We know that

$|\vec{R_{\infty}}| = \sqrt{|\vec{r_{1}}|^{2}+|\vec{r_{2}}|^{2}+|\vec{r_{3}}|^{2}+\ldots} = |\vec{r_{1}}|\sqrt{\sum_{k=1}^{\infty} \frac{1}{k^{2}}} = \pi$

Since we know $\displaystyle |\vec{r_{1}}|\sqrt{\sum_{k=1}^{\infty} \frac{1}{k^{2}}} = |\vec{r_{1}}|\sqrt{\frac{\pi^{2}}{6}} = \pi$ , it follows that

$\vec{r_{1}} \cdot \vec{r_{1}} = |\vec{r_{1}}|^{2} = \boxed{6}$

When the problem says $\vec{R_{\infty}} = \pi$ , it should say $|\vec{R_{\infty}}| = \pi$ .

Jon Haussmann - 6 years, 4 months ago

Hi , good question , should get at least a Level 4 rating . And thanks for the proof of $\sum_{k=1}^{\infty} \frac{1}{k^{2}} = \frac{\pi^{2}}{6}$ , as I used wolfram Alpha to evaluate the sum .

Also I have a doubt , what's the difference between using () and [] in LaTEX ?

Thanks for the same .

A Former Brilliant Member - 6 years, 4 months ago

() does not center your LaTeX, but [] does (and creates a new line for the maths). The former is useful for inserting just small, not-so-important statements/terms/etc, whereas the latter shows your steps more clearly.

Jake Lai - 6 years, 4 months ago

Thanks and keep posting such good questions .

A Former Brilliant Member - 6 years, 4 months ago

@A Former Brilliant Member – You're welcome! Happy solving.

Jake Lai - 6 years, 4 months ago

kinda overrated now...

Julian Poon - 6 years, 4 months ago

In retrospect, I should have made this a level 4.

Jake Lai - 6 years, 4 months ago

Good use of the Basel Problem.

Aakarshit Uppal - 6 years, 4 months ago

Magnitude

The answer is 6.

1 solution

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