- #1
Rosicky08
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WHY there is no scalar equation for a line in three dimensions ?
WHY there is no scalar equation for a line in three dimensions ?
- Thread starter Rosicky08
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In summary, there is no scalar equation for a line in three dimensions because a single equation would only describe a two-dimensional surface, leaving one dimension unaccounted for. To accurately describe a line in three dimensions, at least two equations are needed.
- #2
coalquay404
- 217
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Perhaps you might like to try rephrasing that in a manner which is understandable.
- #3
christianjb
- 529
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How long is a piece of string?
- #4
pmb_phy
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There isRosicky08 said:
ds2 = [itex]\eta_{\alpha\beta}[/itex] dx[itex]\alpha[/itex] dx[itex]\beta[/itex]
Pete
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- #5
masudr
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christianjb said:
Forty two.
- #6
christianjb
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masudr said:
All pieces of string are of length 42, just in different units.
- #7
HallsofIvy
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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 anyone 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
pmb_phy
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Well said. :rofl:masudr said:
1. Why can't a line in three dimensions be represented by a simple scalar equation?
A scalar equation is an equation that only involves scalar quantities, or those that have magnitude but no direction. A line in three dimensions has both magnitude (length) and direction, so it cannot be fully represented by a scalar equation.
2. Can't we just use a scalar equation with three variables to represent a line in three dimensions?
While we can use three scalar variables to represent a line in three dimensions, this would not be a single scalar equation. The equation would still have multiple variables, each representing a different dimension, and would not be considered a scalar equation.
3. Are there any other types of equations that can represent a line in three dimensions?
Yes, there are other types of equations that can represent a line in three dimensions, such as parametric and vector equations. These equations involve both magnitude and direction, making them better suited for representing a line in three dimensions.
4. Why do we need to use more complex equations to represent a line in three dimensions?
Using more complex equations allows us to fully represent the characteristics of a line in three dimensions, including its direction and position in space. Scalar equations are limited in their ability to do so, making them insufficient for representing a line in three dimensions.
5. Is there a specific reason why there is no scalar equation for a line in three dimensions?
The main reason is that a scalar equation only involves quantities with magnitude, while a line in three dimensions also has direction. This means that a scalar equation alone would not be able to fully describe a line in three dimensions, making it an inadequate representation.
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