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This question already has an answer here:

I was thinking about the unit of measure of the Energy (as well as Heat and Work) which is Joule

$$\text{J} = \frac{\text{kg}\cdot \text{m}^2}{\text{s}^2} = \text{N}\cdot \text{m}$$

But again, watching the very last expression I realized that that is the same unit of measure of a vector quantity, namely the Torque or the Moment of Force.

How is that possibile, since the Joule represents a scalar quantity?

Isn't it a bit ambiguous?

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marked as duplicate by Les Adieux, Jon Custer, user36790, ACuriousMind Oct 18 '16 at 0:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ 2 physical quantities a and b will obviously give the same units for s = $(a\cross B)$ and t = $(a . b)$. but one will be a vector and the other a scalar, having completely different meanings. $\endgroup$ – Lelouch Oct 17 '16 at 9:49
  • $\begingroup$ a and b being 2 vectors. $\endgroup$ – Lelouch Oct 17 '16 at 9:51
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There's nothing about units that makes them associated with scalars or vectors specifically. The difference between a scalar and a vector is the fact that the latter has components, but in either case, they can have any units.

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One is defined via a dot product, work $W = \vec F \cdot \vec r$ which produces a scalar quantity, and the other via a cross product, torque $\vec \tau = \vec r \times \vec F$ which produces a vector quantity.

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Since work done $W$ is the sum of force $F$ applied in the direction of displacement $s$, i.e: $$W=\int F\cdot ds $$ we have work as a scalar quantity. Besides, unit defines a magnitude of a quantity, not whether something is a vector or scalar.

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