Coordinate of a point
Extending the notion of coordinates beyond distanceThe above definition designates distance to the number, i.e. . However, if the dimension of the axis is time, the number for the point P will mean . Similarly, for temperature axis it will mean . And so on ...
Therefore, the magnitude of any scalar quantity in terms of coordinate can be generalized as
Magnitude (i.e coordinate of magnitude) change when dimensional unit changes.
Problem setup and statement
For simplicity consider the[M], [L], and [T] such that,
- where a, b, c ∈ ℜ
- of the entity when the dimensions mass, length, and time are measured in certain units; let us refer to them as "original units".
Imagine introducting new units such that,
- the orginal to new unit relation is
- where, A, B, and C are arbitrary positive constants ∈ ℜ+.
- x̄ is the coordinate of the magnitude in terms of the new units.
Solution: Algebraic formula for transformation
Let us use the notation
- 1 OM = A ⋅ NM
- 1 OL = B ⋅ NL
- 1 OT = C ⋅ NT
Multiplying both sides by [(NM)a ⋅ (NL)b ⋅ (NT)c]−1 we get,
Generalizing this for the seven fundamental dimensions [M], [L], [T], [A], [K], [cd], and [mol] we have
Eg.1: Given, 900 ft/min2 what is the coordinate of magnitude for the changed units meter/sec2?
Since, the original unit is ft/min2
For the new unit meter/sec2 we know that
- 1 ft = 0.3048 ⋅ meter
- 1 min = 60 ⋅ sec
To find the coordinate of magnitude for the changed unit