This is somewhat a basic question about vectors since I'm just learning about them. I'm having some confusion.
Say that we have a vector (1,1) and it thus has a magnitude of square root of 2. I realize this is calculated by the pythagorean theorum. Throughout all my years of math I've always used this kind of application for finding distances between lines. However, with vectors it seems the magnitude is something that can be many different things, including force.
My confusion comes when I think about relating force to the coordinates given. If we're talking about force, does that mean that the coordinates given are somehow quantities on some sort of plane that relates to force rather than distance? I'm confused about how the coordinates relate to anything else but distance.
Thank you!
The key here is that force is not just a number. Force is properly described as a number with a direction; i.e. a vector. Example: let's say I'm pushing on a box. Then I'm certainly exerting some force on it; let's say $5$ Newtons of force. Does that number give you all the information you need to be able to describe the effect my force has on the box (assuming you know all the stuff like the mass of the box and the coefficient of friction between the box and the floor, etc)? No. You also need to know in which direction I'm pushing. But if I told you I was exerting a force of $5$ Newtons to the East, then you would have all the information you need.
Let $\mathbf i = (1,0)$ represent a force with a magnitude of $1\ N$ pointing directly to the East. Likewise let $\mathbf j=(0,1)$ represent a force with a magnitude of $1\ N$ pointing directly to the North.
Then let's consider the force $(1,1) = \mathbf i + \mathbf j$.
Using the Pythagorean theorem we see that the magnitude of this force is $\sqrt{2}$. But that's not the only information encoded in this tuple. Notice the direction that the vector in the image points -- that's the direction of the force. In this case the direction has equal parts to the North and East -- thus it points $45^\circ$ East of North.
So $(1,1)$ is the tuple representing the force with magnitude $\sqrt{2}\ N$ that points $45^\circ$ East of North.