However, one thing that does confuse students, especially since it is given in the problem, is why the length of the ropes do not matter in the problem.
Take the problem given above and imagine you were a small atom in the rope on the left.
Strictly speaking though, tension is not the same thing as a force because it has no direction (but does have magnitude).
But we will not need to get into the nuances of what tension is or is not for these problems.
[mathjax] Many of my students find tension in rope problems to be rather difficult.
This does not come at a surprise because there are many moving parts to these problems: the strange phenomenon of tension, basic Physics knowledge, breaking vectors into components, solving simultaneous equations, and some usually not so fun numbers.
This 'reactionary' pullback force is tension in the rope. [Notice the rope feels this force but does not exert it in a direction.
The rope is not pulling, so there is no true direction to this, hence why tension is not a true force.] This is all one needs to know about tension to understand how to do the problems involving tension in Calculus III.
Now it should be easy for us to find the horizontal and vertical components of this vector!
Note that what we are about to do will work only with the magnitude of these vectors, i.e. This is because we want to work with the sides of the triangles using Trigonometry and for that you need to use lengths, not vectors. But what if the problem asks not for the tensions in the rope but asks to find them as vectors?