Is the point $(3, 2, −1, 4, 1)$ in the open ball $B^{5}$ $((1, 2, −4, 2, 3), 3)?$

Question

This is question 11 from the text here.

Is the point $(3, 2, −1, 4, 1)$ in the open ball $B^5 ((1, 2, −4, 2, 3), 3)$?

In my attempt to solve this question, I took the point, subtracted each corresponding component from the ball's center, and then summed the squares of these values: $$(3-1)^2+(2-2)^2+(-1-(-4))^2+(4-2)^2+(1-3)^2=21$$

Because $21>r^2$, that is $21>9$, I concluded that the point is not in the open ball. However, the correct answer is yes, the point is inside the open ball. (Answers are posted here)

For reference (because it might help explain my confusion) this is the definition of an open ball from the book:

The set of all points $(x_1, x_2,...,x_n )$ in $\mathbb{R}^n$ which satisfy the inequality $(x_1 − p_1 )^2 + (x_2 − p_2)^2 +...+ (x_n − p_n )^2 < r^2$ (1.1.15) is called an open n-dimensional ball with radius r and center p, which we denote $B^n(\textbf{p},r)$.

Can someone help me figure out where I am going wrong? Thanks!

Update: I contacted the author of this textbook, Dan Sloughter, and he promptly corrected the answer guide. Also, I was using an outdated version of the book/answers. In case anyone needs them in the future, the updated links are: http://www.synechism.org/wp/the-calculus-of-functions-of-several-variables/ and http://dananne.org/dw/doku.php?id=cfsv:cfsv

You seem to be correct. Maybe there is a mistake in those answers? — Dennis Gulko, Jul 09 '13 at 17:09
This also looks correct to me. My suspicion is that the 'yes' is supposed to be to question 10 (where the point is in the open ball but it would be easy to think it wasn't), not question 11. — Steven Stadnicki, Jul 09 '13 at 17:12
You will note that the page with the answers contains a comment about another mistake in that same problem set. — vadim123, Jul 09 '13 at 18:13

amWhy · Accepted Answer · 2013-07-09T18:43:33.317

Your definition of the Euclidean metric, your work, and your conclusion are all correct. Based on that metric, the given point is not in the open ball $B^5 ((1, 2, −4, 2, 3), 3)$.

I think you can trust your work.

I suggest you post a comment (as requested by the "author" of the answers) stating that there is an error in the answer to $(11)$.

Additional suggestion: In that same comment, post a link to the URL for you question here at math.se, to show both the correct answer, and how your correct solution is obtained. That way, the solution can be corrected by its author, while at the same time and until the correction, help alert anyone else who is perplexed by that same question.

always excellent suggestions. It has been a really slow day! +1 — Amzoti, Jul 10 '13 at 00:08

score 1 · Answer 2 · answered Jul 09 '13 at 17:17

Your calculation looks fine.

Due to the distance (with respect to the euclidian metric) in the 3rd coordinate is equals to 3, the point cannot be in the open ball with radius 3.

So either there is given a different metric or the given solution is simply wrong.

Is the point $(3, 2, −1, 4, 1)$ in the open ball $B^{5}$ $((1, 2, −4, 2, 3), 3)?$

2 Answers2