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I was thinking about the way our system of units of work, and I realized that we have been multiplying units (such as for lengths) by negative numbers, when dealing with vector quantities; positive constants being multiplied by a meter makes sense - 5 m is m scaled up by 5. So what do we mean when a vector is this?

$$ \begin{pmatrix} -3m\\ -4m\\ \end{pmatrix} $$

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Based on what your axes are and which direction along those axes you define a negative direction to be, you are $3m$ in the negative direction along the first axis, and $4m$ in the negative direction along the second axis.

Another interpretation: you have two axes defined. On each axis you have put a number line that contains all of the real numbers in terms of meters. Your vector's first component matches up with where $-3m$ is on the first axis, and its second component matches up with where $-4m$ is on the second axis. This interpretation makes "defining a negative direction" more explicit than in the one above.

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