By Frederick S. And Frederick H. Bailey Woods
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An additional important property that can be proven using the three defining properties of an inner product given above is the Schwarz inequality: |a · b| ≤ a b ∀ a, b ∈ Rn d . 16) v= v v where ev is the unit vector in the direction of v. For example, if v is the velocity vector, v is the magnitude of the velocity (absolute speed) and ev is the direction of motion. 1, we stated that positions are measured relative to some specified physical object. However, the actual act of measurement requires the definition of a coordinate system – a standardized scheme that assigns a unique set of real numbers, the “coordinates,” to each position.
Albert Einstein was intrigued by Mach’s thinking, but the conclusion to emerge from the special theory of relativity was that in fact there would be tension in the rope even in an empty universe [Gre04, p. 51]. t Scalars, vectors and tensors 12 Relative space is some movable dimension or measure of the absolute spaces, which our senses determine by its position to bodies and which is commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth.
For this reason the modern definition of inertial frames does not refer to absolute space, but instead relies on Thomson’s law of inertia, which is described shortly. 8 9 Many books on mechanics take the view that Newton’s laws only hold for systems of point particles that by definition have constant mass. In this case, conservation of mass is trivially satisfied and need not be mentioned. The view presented here is more general and consistent with the generalization of Newton’s laws to continuum systems which is adopted in the later chapters.