What Geometric Shapes Do Equations Represent in Higher Dimensions?

[snipez90]

Homework Statement

What does the equation $$x^2 -2x + y^2 -2y + z^2 -2z + t^2 -2t + 4 = 0$$
represent in $$\Re^4$$? What does it represent in $$\Re^5$$? What does it represent in $$\Re^6$$?

Homework Equations

Complete the square.

The Attempt at a Solution

After completing the square (letting 4 = 1 + 1 + 1 + 1), the equation becomes

$$(x-1)^2 + (y-1)^2 + (z-1)^2 + (t-1)^2 = 0$$

In $$\Re^4$$, the equation represents the coordinate (1,1,1,1). I have a hard time visualizing the cases for $$\Re^5$$ and $$\Re^6$$. I figure that it should be easier to visualize knowing that in $$\Re^4$$, we have a fixed point.

Of course I have deduced that we will get a line and a plane in $$\Re^5$$ and $$\Re^6$$, respectively, by looking at smaller cases. But what is the exact configuration and how do I see it?

 

[tiny-tim]

… don't worry …

What does it represent in $$\Re^5$$? What does it represent in $$\Re^6$$?
…
Of course I have deduced that we will get a line and a plane in $$\Re^5$$ and $$\Re^6$$, respectively, by looking at smaller cases. But what is the exact configuration and how do I see it?

Hi snipez90!

"a line and a plane" is the answer!

If you want to give the exact equation, the line, for example, would be x=y=z=t=1 (and w = anything).