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I'm not sure what to make of the direction of a vector with components (0,0,0). Is it an undefined vector?

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There's absolutely nothing wrong with a vector having all zero components. But a vector like that does have an undefined direction. So the definition of a vector as "something which has a magnitude and a direction" can throw you off here a little.

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In mathematics that is the identity element 0 for vector addition, and is a key object in the definition of a vector space.

Physically, if $(0, 0, 0)$ is the position vector of some particle, then that just means it is located at the origin.

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