# WHY there is no scalar equation for a line in three dimensions ?

1. Apr 13, 2007

### Rosicky08

WHY there is no scalar equation for a line in three dimensions ???

2. Apr 13, 2007

### coalquay404

Perhaps you might like to try rephrasing that in a manner which is understandable.

3. Apr 13, 2007

### christianjb

How long is a piece of string?

4. Apr 13, 2007

### pmb_phy

There is

ds2 = $\eta_{\alpha\beta}$ dx$\alpha$ dx$\beta$

Pete

Last edited: Apr 13, 2007
5. Apr 18, 2007

### masudr

Forty two.

6. Apr 18, 2007

### christianjb

All pieces of string are of length 42, just in different units.

7. Apr 18, 2007

### HallsofIvy

Rosicky08, R3 has three dimensions. A curve has one dimension so you have to "lose" 2 dimensions. If you had a single, scalar equation, in three variables, you could solve for any one in terms of the other TWO. That gives you "two degrees of freedom": pick any two coordinates and solve for the third: that one equation allows you to "lose" on dimension. Any single scalar equation describes a two dimensional surface. In order to get one dimension you must have either 2 equations in the three space coordinates so that you could solve for two variables in terms of 1 or 3 equations in the 3 space coordinates and one parameter: 3 equations in 4 variables gives you 4-3= 1 degree of freedom or one dimension.

8. Apr 18, 2007

### pmb_phy

Well said. :rofl: