Operation of Units in Vector Quantities

Question

If we consider torque as a cross product of the force and radius vectors (pointing outwards). $$\overrightarrow{} =\overrightarrow{r} \times \overrightarrow{F}$$

Where the force is measured in newtons and the radius in metres.

And work:

$$W =\overrightarrow{F}\cdot \overrightarrow{s}$$

Where the force is in newtons and displacement in metres.

Torque is from the cross product of radius and force, two vectors, and therefore creating another vector. The radius is treated like a vector much like area in electric and magnetic surface integrals.

Work is the dot product, where a scalar quantity is made from two vectors.

In both cases, the units are simply multiplied together, however joules is commonly used for work, and newton metres for torque, although this could describe work as well. Therefore, what are the differences between units as they describe terms in cross and dot products? What are the physical implications and differences?

Answer 1

Usually the newton-meter is reserved for torque and the joule is reserved for energy so as to avoid confusion. Although torque and work both multiply force and distance, the type of product--cross vs dot--affects the interpretation of these units. This question answers the physical difference quite well. Torque's $N\cdot m$ implies an influence in space with a particular direction, whereas joules (for work) indicate a completed action with energy transferred. Physically torque applied without rotation doesn't accomplish work. It just applies force at a distance without moving anything.