Frequently Made Errors in Vectors - Elementary Use - Comments

[haruspex]

haruspex submitted a new PF Insights post

Frequently Made Errors in Vectors - Elementary Use

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[Orodruin]

Generally good insight as usual, but I feel I must nitpick a bit here, since you are mentioning the issue of vectors belonging to a position:

Some physical entities that behave as vectors are not completely characterised by their vectors. A force not only has magnitude and direction but also a point of application (a second vector). On the other hand, position, velocity, acceleration, and further time derivatives are each completely described by their vectors.

While this may be true for the most basic use of vectors in terms of vectors in a Euclidean space, it is not generally true. For the more general vector concept, every vector is associated with a position and belongs to the tangent (or cotangent) space at that position.

Position is generally not a vector. Velocity belongs to the tangent space at the position of the object, as does acceleration. In a Euclidean space the position of the velocity vector may often be ignored as parallel transport is trivial, but consider movement on a sphere (such as the Earth's surface) for example.

 

[Greg Bernhardt]

What an incredible resource this series has become!