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As a physicist myself, I find I think of vectors more often as a set of independent magnitudes for each of the coordinate system axes, than of a single magnitud

As a physicist myself, I find I think of vectors more often as a set of independent magnitudes for each of the coordinate system axes, than of a single magnitude with a direction. That's because in most cases, the physics just works independently in each axis. So it makes a lot of sense to treat vectors as sets of numbers. Most calculations physicists do are the same as pure mathematicians would do. So we use for example in-product and cross-product operators to operate on vectors, without caring for the exact coordinate system used. Only when we have to come up with a final answer to a question like "what angle does the ball hit the ground and with what velocity?" will we convert our vectors into a magnitude and direction.

jacobolus 7y ago
> I find I think of vectors more often as a set of independent magnitudes for each of the coordinate system axes Interesting. This is what Hestenes calls the “coordinate virus”, http://geocalc.clas.asu.edu/pdf/MathViruses.pdf http://geocalc.clas.asu.edu/pdf/MathViruses.pdf ; very often any specific coordinate system is not inherent but is some arbitrary addition to the space made for convenience in some particular calculation. It is in my opinion a mistake to think of the coordinates as primary. > operate on vectors, without caring for the exact coordinate system used This is the opposite of the previous statement.
- gsliepen 7y ago
  > This is the opposite of the previous statement. Not really. I didn't say in the first statement that a specific coordinate system was used. In most systems, it doesn't matter how you choose your coordinate system, as long as the basis is orthogonal, treating the coordinates independently of each other will work.