# Why is work a scalar?

..and I don't know what else to use to show that work is a scalar.

Now this is going in circles, it's a scalar because it's defined to be a scalar. [gives up]

Wikipedia, on the work-energy theorem:
The theorem is particularly simple to prove for a constant force acting in the direction of motion along a straight line. For more complex cases, however, it can be claimed that very concept of work is defined in such a way that the work-energy theorem remains valid.

Why insist on understanding work without any reference to energy? Energy is the more fundamental quantity.

Ok, all I can say is

1) Dot product of two vectors is scalar, thus work is scalar.

2) Work is not vector; you could not write it as Wx+Wy+Wz because IT ISN'T VECTOR.

3) Yes...is that clear

jtbell
Mentor
Note that there is a motion-related quantity that has direction: momentum, $\vec p = m \vec v$.

In the 1700s, there was a lot of confusion about whether the "proper" quantity that combines mass and velocity contains v (momentum) or v^2 (kinetic energy). Finally physicists realized that we need both of them, for different purposes.

We define work the way we do, so as to have a conserved quantity via the work-energy theorem:

$$\int{\vec F \cdot d \vec s} = W = \Delta K = \frac{1}{2} mv_f^2 - \frac{1}{2} mv_i^2$$

When we integrate force over time (instead of over distance) we get the change in momentum, via the impuse-momentum theorem:

$$\int{\vec F dt} = \vec I = \Delta \vec p = m \vec p_f - m \vec p_i$$

Delta2
Homework Helper
Gold Member
@JeffKoch

I understand what you're saying, but...
In The Feynman Lectures on Physics, Volume I, an exploration of torque produces a motivation for the definition of the vector (cross) product. I think there is a similiar procedure with work, but I don't know why work is defined as a scalar. The work-energy theorem in one dimension is not useful for this, and I don't know what else to use to show that work is a scalar.

So you asking why work is scalar and why dot product is a scalar? Hard to resolve this without falling back to the definition of energy and that energy is a scalar (therefore it doesnt matter if we have kinetic energy on the x,y or z axis we just add em to find the total energy). If work wasnt a scalar how it could be a vector what physical meaning could you give to the direction a work could have in order to be a vector?

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