Difference between projection and component of a vector in product?

Question

This is a very basic question about dot products or scalar products:-

If I want to move a block and I apply a force parallel to displacement, the block will move and some work will be done. So in the formula will be $W= F\cdot S$, here we won't calculate the force mg of the block but the force we applied (the parallel force).

Now let's say that the force is not parallel and is at some angle from the horizontal

So in this case the work done will be the projection of the force $F_1$ on the $x$ axis, because that is how dot products are defined (As Projections). But can't we say that the block moved due to its horizontal component $F_1\cos(\theta)$ and the answer would be same. And obviously we won't count $F_1\sin(\theta)$ as the work is not done by it.

So why do we say projection and not component in dot or vector product ?

Answer 1

As pointed out, the projection and component actually refers to the same thing. To solve a problem like this it useful to introduce a coordinate system, as you mentioned yourself you project onto the x-axis. As soon as you introduce a coordinate system you can talk about the $\textit{components}$ of some vector. E.g $\vec{F} = F_1 \cos(\theta)\hat{x} + F_1 \sin(\theta) \hat{y}$ and $\vec{S} = S_1 \hat{x}$ if the box in your example is constrained to move horizontally. Here the components of the vectors are the $\textit{projections}$ of the vectors onto the coordinate axis. With this construct you calculate the dot product as $W = \vec{F} \cdot\vec{S} = F_1\cos(\theta) S_1$.

However it is not needed to introduce a coordinate system and writing the vectors by their components and then applying the rules for dot product. The dot product is defined in a coordinate independent way as a projection. So your question is just a matter of terminology. A $\textit{component}$ of a vector along some axis is the $\textit{projection}$ of the vector along that axis and in this sense projection is the more fundamental thing.

Answer 2

Projection and component mean the same thing in this context.

Let me however note that a force is a part of the physical reality, existing in the world (at least for the purposes of mechanics), whereas a projection is a mathematical object/construct. From this point of view the work is done by the force (or by its component), but not by the projection.