Work as a Dot Product

Question

Why is

Work $=$ Newtons $\times \cos (\theta) \times$ Distance

and not

Work $=$ Newtons $\times \cos(\theta)$

My understanding is that Work is Newtons directed horizontally so why not just project the Newtons of the original force in the horizontal direction. Why multiply by the magnitude of the distance vector? (Assuming we don't already know it's a Dot product).

Answer 1

That's just what the work formula is: $\text{work = force $\times$ distance}$.

The force multiplied by $\cos\theta$ just gives you the horizontally directed piece of the force. It doesn't tell you how far that piece is directed, i.e., it doesn't tell you the distance. You need to multiply by the magnitude of the vector to get this distance.

My understanding is that Work is Newtons directed horizontally so why not just project the Newtons of the original force in the horizontal direction.

Work is not necessarily in the horizontal direction. Work can be done in any direction.