# Why is work scalar and the dot product of force and displacement?

I asked many people why work is scalar. But the questions and the answers just cycles.

My question : Why is work a scalar quantity?

Their answer : Because it is the dot product of Force and Displacement. So it is a scalar quantity.

So I asked,

My question : Why is work the dot product of force and displacement and not the cross product?

Their answer : Because it is a scalar quantity and not a vector.

So can anyone please tell me why work is scalar apart from this cycle of questions and answers? Is there any other logic which can answer both questions?

Edit : $W=Fs\cos\theta$. So $Fs\cos0 = -Fs\cos180$. So shouldn't work be a vector? This confusess me.

• If you are happy with my answer, you can accept it. Otherwise let me know in a comment under my answer if you still have doubts and I can address them. – Andrea Aug 1 '18 at 11:10

## 1 Answer

Well, say that you are looking at a man lifting boxes. Each box weighs 10kg. At first you look at him standing, but since just looking at him made you tired, you decide to lie down. Now, from your horizontal position, the scene looks different, but is the man found more, less, or the same amount of work per box?

The word “scalar” is often used to just mean “number”, but it actually has a more technical word. A scalar is a quantity that does not change when the system is transformed following one of the symmetries of the theory.

In this case, I think you are talking about classical mechanics, and the transformations involved are rotations in 3D space. As you know, the dot product of two vectors is invariant under rotations (that is probably why they also call it “the scalar product”).

So here it is. Work should not vary if you look from different perspectives, so you want it to be a scalar. Also, in any specific frame, work should be force times displacement, like you learn in school. So the best way is to define works as the scalar product of the force and the displacement vectors.

• Oh sorry. I totally forgot about that. – Theoretical Aug 1 '18 at 11:31
• @AsifIqubal no worries ;) – Andrea Aug 2 '18 at 7:06