resultant of forces
When several forces push at one point, their effect can be replaced by a single force, and that one force is the resultant. Replacing many forces by one resultant tidies the picture down to a single arrow. You can join arrows head to tail, draw a parallelogram, or split each force into components and add the horizontals together and the verticals together. The three methods agree. Start by turning the second force and watching the resultant follow.
The intuitive way to add forces is to join the arrows. Draw the first force F1 and start the tail of the second force F2 at its head, and the head of F2 is where the two forces add. The arrow drawn straight from the start to that last head is the resultant. Turn F2 and wherever F2 points the resultant still runs from the start to the last head. With more forces you keep joining head to tail and connect the first point to the last, and changing the order does not change the arrival.
With two forces there is a tidier picture. Start both forces at the same point and draw a parallelogram. Take F1 and F2 as the two sides and the diagonal that crosses between them is the resultant. This is the same story as joining head to tail. The opposite side of the parallelogram is F2 slid parallel, which is the same as attaching F2 to the head of F1. Turn F2 and the parallelogram squashes while the length and direction of the diagonal follow, growing when the two forces point the same way and shrinking when they oppose.
A picture is fine for two forces, but when the forces grow it is a nuisance to measure with a ruler, so in practice you add components. Split each force into horizontal and vertical, add the horizontals together and the verticals together. Here three forces meet at a point, and turning the third force F3 changes the two sums at once. Mind the signs: right and up are positive, left and down are negative. The two numbers you gather are the horizontal and vertical of the resultant, so however many forces there are they shrink to two numbers.
From the gathered horizontal and vertical, find the size and direction of the resultant. The hypotenuse of the right triangle with those two sides is the resultant R, so the size is R = √(ΣFx² + ΣFy²) and the direction is θ = arctan(ΣFyΣFx). Turn F3 and the three forces have gathered into one thick arrow, the resultant R. When you read the direction, check the quadrant from the signs of the two components, because trusting the arctangent value alone makes it easy to point the opposite way. Several forces tidy into one size and one angle.
Turn the direction around. If composition gathers several forces into one, resolution opens one into several, and the two are the two ends of the same relation. You can split one force back into any two directions you choose, and they need not be horizontal and vertical. There is a resultant and two slanted guide lines, and if you resolve the resultant along those directions and join the pieces head to tail you return exactly to the resultant. Change the angle of the guides and the sizes of the two pieces change, but they still add to the same resultant. This reverse is needed when a force must be split along a slope or a cable.