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Let's say for example I have these three vectors representing these three fields:

$$\begin{array}{c|c|c} \text{Cars Sold} & \text{Cars in Lot} & \text{Profit Generated}\\ \hline \\140 & 89 & 987781 \\ 23 & 210 & 789000 \\ 300 & 12 & 1900000 \end{array}$$

I want to find which vector is the most similar to this: $[189, 45, 1300000]$

Since these vectors have fields that aren't all of the same magnitude, would it be appropriate to use euclidian distance between these vectors to find which has the shortest distance?

Also, could cosine similarity be used as well?

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You could do either, but you might not get what you want. The Euclidean distance will be dominated by profit, as the range is so much larger. If you measure profit in 1000s or 10000s you will see the other components more. Philosophically, the vector you pick as most similar shouldn't depend on scaling.

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