Exploring the Unit Cube in [tex]\ R^{n}[/itex]

[shwin]

I'm having trouble visualizing [tex]\ R^{4}[/itex](a domain of reals in four dimensions).

1. Describe a procedure in given 3 vectors, finds a fourth vector perpendicular to those three. Explain why we can use it in analogous fashion to the normal vector to a plane in [tex]\ R^{3}[/itex].

Here, I'm thinking taking the normal vector of each vector and then adding the three normals would be sufficient, but I am not sure how this is analogous to the normal vector to a plane in the xyz system.

2. How many vertices does the unit cube have in [tex]\ R^{n}[/itex] have? What is the furthest distance from the origin that one can be on the unit cube in [tex]\ R^{n}[/itex]? What is the average distance of the vertices to the origin?

First part I have [tex]\ 2^{n}[/itex], second part a square root of [ a summation from i = 1 to i = n of [tex]\ n^{2}[/itex]]

But the last part stumps me...average distance? and writing a formula for this is a bit confusing too.

 

[mrandersdk]

for the first you should look at the gram-schmidt proces:

http://en.wikipedia.org/wiki/Gram-Schmidt_process

for the second, you are right about the $$2^n$$. In general the distance from a vertice to the origin is given by

$$\frac{\sqrt{1^2+1^2+\dots+1^2}}{2} = \frac{\sqrt{n}}{2}$$

so the average must be

$$lim_{n\rightarrow \infty}\frac{1}{n}\sum_i^{n} \frac{\sqrt{i}}{2} = lim_{n\rightarrow \infty} \frac{1}{n}[\frac{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}{2}] = lim_{n\rightarrow \infty} \frac{\sqrt{1}}{2n}+\frac{\sqrt{2}}{2n}+\dots +\frac{\sqrt{n}}{2n} = \infty$$

edited from:

$$lim_{n\rightarrow \infty}\frac{1}{n}\sum_i^{n} \frac{\sqrt{i}}{2} = lim_{n\rightarrow \infty} \frac{1}{n}[\frac{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}{2}] = lim_{n\rightarrow \infty} \frac{\sqrt{1}}{2n}+\frac{\sqrt{2}}{2n}+\dots +\frac{\sqrt{n}}{2n} = 0$$

if the average is over dimension.

 

[mrandersdk]

ups, don't think the limit converge

 

[mrandersdk]

then I'm not sure what average you should calculate

 

[HallsofIvy]

I'm having trouble visualizing [tex]\ R^{4}[/itex](a domain of reals in four dimensions).

1. Describe a procedure in given 3 vectors, finds a fourth vector perpendicular to those three. Explain why we can use it in analogous fashion to the normal vector to a plane in [tex]\ R^{3}[/itex].

Here, I'm thinking taking the normal vector of each vector and then adding the three normals would be sufficient, but I am not sure how this is analogous to the normal vector to a plane in the xyz system.

There is no such thing as "the" normal vector to a vector- any vector in the 3 dimensional "plane" normal to that vector. And even in 3 dimensions, adding normal vectors to two given vectors will not in general give you a vector normal to both. In 3 dimensions you have to take the cross product of the two given vectors. Can you thknk of a procedure analagous to that?

2. How many vertices does the unit cube have in [tex]\ R^{n}[/itex] have? What is the furthest distance from the origin that one can be on the unit cube in [tex]\ R^{n}[/itex]? What is the average distance of the vertices to the origin?

First part I have [tex]\ 2^{n}[/itex], second part a square root of [ a summation from i = 1 to i = n of [tex]\ n^{2}[/itex]]

But the last part stumps me...average distance? and writing a formula for this is a bit confusing too.

Assuming by "unit cube" they mean an n-dimensional "cube" with center at the origin and each side of length 1, then you have, NOT a " summation from i = 1 to i = n of n^2" but only a summation from 1 to n of 1 (the square of the side length= 1)- and that is just n. The diagonal distance between opposite vertices is $\sqrt{n}$ and if the origin is at the center of the "cube" the distance you want is $\sqrt{n}/2$ (I see that mrandersdk has already given that). Now I guess you could divide that by n and sum to get an average but that does not converge!