# Over the definition of Joule in relation to a vector quantity [duplicate]

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?

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

• 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. – Lelouch Oct 17 '16 at 9:49
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.
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.