Many Forces Gather into a Single Resultant
What happens when several forces act on one point at the same time? Happily, their whole effect can be replaced by just one force. That one is the resultant. Turning many forces into a single resultant cleans a busy picture down to one arrow. There are two pictures and one calculation for combining them: lay the arrows head to tail, or draw a parallelogram, or split each into components and add the horizontals with the horizontals and the verticals with the verticals. All three give the same answer. Start by turning the second force yourself and watch how the resultant follows.
The most intuitive way to add forces is to chain the arrows. Draw the first force F1, and start the tail of the second force F2 at its head. Then the head of F2 marks where the two forces add up to. The arrow drawn straight from the very first tail to the very last head is the resultant R. Turn F2. Wherever F2 points, R always runs from the start to the last head. With three or four forces it is the same: tail to head, tail to head, and then join the very first to the very last. Change the order and you still arrive at the same point.
For two forces there is an even cleaner picture. Start both forces from the same point and draw a parallelogram from the two arrows. Take F1 and F2 as the two sides, and the diagonal that crosses between them is the resultant R. This is really the same story as the tip-to-tail method: the opposite side of the parallelogram is just F2 shifted over, so it is the same as sticking F2 onto the head of F1. Turn F2 and the parallelogram skews, and the length and direction of the diagonal R follow. The more the two forces point the same way, the longer R; point them opposite and they cancel down to a shorter R.
Pictures are fine for two forces, but with three or four, measuring with a ruler gets clumsy. So in practice we add by components. Split every force into a horizontal and a vertical part, then add all the horizontals into ΣFx and all the verticals into ΣFy. Here three forces act at one point. Turn F3 and the two running totals ΣFx and ΣFy change live. You must keep the signs: right and up are positive, left and down are negative. These two collected numbers, ΣFx and ΣFy, are the horizontal and vertical components of the resultant. However many forces there are, it all shrinks to just two numbers.
Now the last step. With the collected ΣFx and ΣFy, find the magnitude and direction of the resultant. The resultant R is the hypotenuse of the right triangle whose sides are ΣFx and ΣFy, so its size is R = √(ΣFx² + ΣFy²) and its direction is θ = arctan(ΣFyΣFx). The three forces from the previous scene are gathered here into one bold arrow, the resultant R. Turn F3 and the length and angle of R change together. When you read the direction, always check which quadrant it is in from the signs of ΣFx and ΣFy; trusting arctan alone makes it easy to point the exact opposite way. This is how many tangled forces settle into one size and one angle.
Finally, let us run it backward. If composition gathers many forces into one, decomposition unravels one into many; they are two ends of the exact same relationship. So you can also resolve a single force back into any two directions you choose, and they need not be horizontal and vertical. Here is a resultant R and two slanted guide lines. Resolve R into components along the two guides, and chaining those two components tip to tail brings you exactly back to R. Change the angle of the guides. The sizes of the two components shift, but together they always make the same R. Whenever a force has to be split along particular directions, like a ramp, a pulley, or a cable, this reverse direction is exactly what you need.