合力作为一个向量

Consider an object $O$ that can only move on one line:

If we wanted to apply a force on this object, we could only apply it in two possible directions: to the left or to the right. Consider force as a signed quantity: a positive force in one direction is a negative force in the other direction.

Now if we extend this definition to two dimensions, we have an object that can only move left and right, up and down. To track an object's position, we project its location onto two perpendicular axes (call this the $x$-axis and the $y$-axis).

In this 2D plane, we can apply forces in infinitely many directions: to the right, to the left, to the upper-right, etc. But, any force we apply on this object in the 2D plane can too be projected onto the $x$-axis and $y$-axis. That is, any force in the 2D plane is the combination of some horizontal force and some vertical force, called the horizontal and vertical components of the force.

If we apply a force of magnitude $1N$ at an angle of $60^\circ$ above the horizontal to an object, then right-triangle trigonometry tells us that the force has a horizontal component of $\left( 1 \, \text{N} \right) \cos60^\circ=\frac12 \, \text{N}$ and a vertical component of $\left( 1 \, \text{N} \right) \sin60^\circ=\frac{\sqrt3}2 \, \text{N}$. Therefore, this force can be represented as a vector, $\left<\frac12 \, \text{N},\frac{\sqrt3}2 \, \text{N}\right>$.

Force is a vector, a quantity with magnitude as well as direction. For example, you can push a box with a certain force to the right, or to the left. You can push with a small amount of force, say 2 Newtons, or you could push with 10 Newtons.

When multiple forces act on an object at once, a net force is developed that accounts for forces that cancel each other out or adds on to each other. This is known as the superposition principle.

There are multiple ways to find out what the net force vector is. A graphical way to approach the problem is the head-to-tail method. Take your first force vector and place its tail (where the vector begins) anywhere, at the origin of the coordinate plane, for example. Take your second vector's tail and put it at the end of the first vector (remember to retain direction in this process!). Repeat this for however many component vectors you may have, and when you are finished, connect the tail of your first vector and the head of your last. That is the net force vector.

You can also break down each component vector in x-components and y-components using a little bit of trigonometry. Find the net x force and net y force by adding and subtracting. You can then use Pythagorean theorem to find the net force vector's magnitude, and a little more trigonometry gets you the direction.