iSoul In the beginning is reality

Dimensions and units

A dimension is informally regarded as the number of coordinates needed to specify the location of a body or point. That may suffice for a mathematical dimension, but a physical dimension is a dimension of something, that is, some unit. In that sense, the dimensions of force are different from the dimensions of velocity.

However, units such as force and velocity use the same dimensions of space and time. They have a common notion of direction as their basis. A northward force and a northward velocity are in the same direction. In that sense, it is common to speak of space and time as the only physical dimensions.

It is possible to use the speed of light to translate spatial units into temporal units and vice versa. This is done in relativity: the invariant interval may be expressed in length units by multiplying time by the speed of light or in time units by dividing lengths by the speed of light. So space and time are integrated as spacetime dimensions.

Outside the technical usage of relativity, length and time units are distinguished because they are physically different. The method of measuring them differs. The common measures of motion are related to time, not space. The philosophy of time is different from the philosophy of space.

For vectors of rates, such as velocity, different component rates lead to different vectors. Consider a displacement/distimement north 90 km in 3 hrs and east 30 km in 2 hours. The rates for each component dimension are 90/3 = 30 km/hr north and 60/2 = 30 km/hr east. However, the time rates use the overall time of 5 hrs and get 90/5 = 18 km/hr north and 60/5 = 12 km/hr east. The space rates use the total distance of 150 km to get 150/3 = 50 km/hr north and 150/2 = 75 km/hr east (which would normally be expressed as hr/km).

That is, for time rates, the dimensions apply to space, and for space rates the dimensions apply to time. For the rates by component, the dimensions apply to velocity itself rather than space or time.

Addendum: If we consider the progress in the direction of the end point in time, the velocity denominator would be the square root of (3²+2²) = √13. The corresponding numerator would be the square root of 90²+30² = √95. The resulting velocity magnitude would be √(95/13) = 7.2 km/hr.

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