force vectors and components
A force on paper is one arrow. The length is the size and the head points the direction, so a force is not one number but a vector that carries size and direction together. A slanted arrow is awkward, so it is split into how much along the horizontal and how much along the vertical. That split is taking components, and it is the moment one slanted force becomes the same as a horizontal force plus a vertical force. This one skill holds up all of statics, because equilibrium and structures are solved in components. It starts by moving the arrow yourself.
Grab the head of the arrow and pull, and the arrow grows as the force grows. Turn the head and the direction changes. Size and direction move apart, and the same size with a different direction is a completely different force. Pushing to the right with 5N and pushing up with 5N give entirely different results. Writing a force means writing the direction along with the size, usually as an angle measured from the horizontal.
Split the slanted arrow into two pieces. How hard it pushes across is the x component Fx, and how hard it pushes up is the y component. Lay the angle down and Fx grows, stand it up and the vertical component grows, and a right triangle appears. The hypotenuse is the force F, the base is Fx, and the height is the vertical component, so Fx = F cosθ, Fy = F sinθ, just the trigonometric ratios. One arrow is rewritten as a horizontal force and a vertical force.
The reverse works too. Set the horizontal force and the vertical force separately and they join back into one slanted force, and moving the sliders shows that joining. Lay the two forces as the sides of a rectangle and the diagonal is the resultant F. The size is Pythagoras, F = √(Fx² + Fy²), and the direction is θ = arctan(FyFx). Splitting takes one into two, and joining takes two into one. When several forces are added, each is split into components, the horizontals are added together and the verticals together, and at the end they are joined back this way.
A structure lives in space rather than in a plane, and the method is the same. Add a depth direction to the horizontal Fx and the vertical. Stack the force along the three edges and the arrow becomes the diagonal of a box, with size F = √(Fx² + Fy² + Fz²). It is the 2D Pythagoras stretched into 3D, so the idea of splitting into components survives the extra dimension. A force in space is still written as the three numbers x·y·z.
Separate the direction alone. Drop the size of the arrow and keep only the direction, and a short arrow of length exactly 1 remains. That is the unit vector u. In the plane u = (cosθ, sinθ) and the length is always |u| = 1, so turning the angle slides the head along the unit circle. The original force is the direction times the size, F = F·u, so u carries the direction and the number in front carries the size. It is the clean way to separate the two pieces of a force, and it is useful whenever the direction is slanted.