Module 4  Net Force and Vector Addition
From PER wiki
It is very common for objects to experience multiple interactions simultaneously. When an object experiences multiple forces their effects will combine with each other, but they cannot simply be added together. The reason is that forces are not just numbers, they have direction associated with them as well. Quantities like force that have both size (called 'magnitude') and direction are called vectors.
Physical Vector vs. Sclar Quantities 

If a vector is a quantity that is defined by a magnitude coupled with a direction, then in keeping with Newton's Second Law it is clear that acceleration is also a vector; the magnitude is the rate of change of velocity, and the direction tells you which direction the velocity vector is going to turn. For example, a dropped rock accelerates in the direction of the ground at about 9.8m/s/s. A projected rock will also be accelerating straight down at about 9.8m/s/s because that is the direction of the net force on it. Velocity, displacement, position, momentum, angular momentum, impulse and torque are other physical vector quantities encountered in the study of mechanics. These quantities all behave the same mathematically. We distinguish physical vectors from physical scalar quantities; scalars have no direction associated, they carry the relative "scale" of the quantity reported in some dimensional system. The inertial mass of an object is a scalar. By example, maybe the mass of your computer is 1.1 kilograms which is equivalent to 0.075 slugs or 1100 grams or 6.62x10^+26 atomic mass units. The numerical value of the mass is different in each of these systems, because the scale varies. There is no other information required to fully quantify the scalar quantity of mass. Other scalar quantities of significance in the study of mechanics are the familiar speed, energy, work, moment of inertia and distance. Avoid ConfusionBecause the magnitude of a vector is a scalar, there is room for confusion when we talk about vector quantities. It is important to state clearly (in words or in symbolic notation) when you refer to the scalar magnitude of a vector or the complete vector quantity. Some vector quantities are so much a part of everyday experience the vector part of the language is not explicit. Common pitfalls when reading, speaking or working in physics are when the casual language of speed, acceleration and force are used to mean the more rigorous vector quantity. "Speed" is the magnitude of the velocity. "g"= 9.8 m/s^2 (often casually called "gravity") is the magnitude of the free fall acceleration due to gravity, straight down to the center of the earth. "weight" is the magnitude of the gravitational force on an object by the entire earth, directed straight down to the center of the earth. As you can see, the language is long and difficult; so to the extent that you shorten up your expressions, be sure you are crystal clear on whether you are referring to a scalar OR just a part of the associated vector OR the vector. Usually you cannot assume your audience knows exactly what you are meaning. Strategies for clarifying this issue on paper: if it is a vector, draw a little dagger arrow over the symbol. If it is a scalar, don't draw the dagger. 
Learning Goals
After completing this module you should be able to:
 Find x and y components of a vector given as a magnitude and direction, and viceversa.
 Add vectors.
 Define the net force acting on an object in terms of the individual forces acting on it.
Forces are Vectors
In the figure below, two people are pushing a heavy crate on a very slippery floor. One of the people pushes with a force of 40 Newtons (in the International System of Units, forces are measured in Newtons, which is abbreviated by the symbol N). The other person pushes with a force of 30 N. It is clear that if the people push in the same direction, the magnitude of the total (or net) force on the crate is 70 N. If the people push in opposite directions, the magnitude of the total force is only 10 N, since they are pushing against each other. If, however, the people push at right angles to one another, the net force is found by constructing a vector triangle. The net force is given by the sum of the force vectors (the hypotenuse of a triangle because these forces are perpendicular), and therefore has a magnitude of 50 N, which is more than either force alone, but less than the sum of their magnitudes.
Summing Arbitrary Vectors
Rule 1: Think Geometrically: Geometric or graphical additions of vectors
Think of vectors as arrows, and add them by putting the tail of one on the arrow of the other as is shown in the two diagrams immediately above. It does not matter which vector's tail is placed on the arrow of the other  the sum is the same (called the commutative property). This geometric addition is very helpful in sketching out preliminary predictions an problems because the direction of F_{net} will always point in the same direction as the acceleration vector. You can draw any vector anywhere on the page, pointed in the correct direction and scaled appropriately, so that the length drawn on your paper corresponds to the magnitude.
 Choose an appropriate scale for length.
 Draw one of the vectors in the sum to scale, in the correct direction.
 Draw the next vector in the sum with the tail of 2 on the head of the first.
 Draw the next vector in the sum with the tail of 3 on the head of 2.
 Keep drawing in this tail of the new on the head of the preceding vector until all the vectors in the sum are drawn.
 Finally, draw the vector that starts at the first tail and ends on the final head, put an arrow on it and measure its length according to the scale.
The resultant vector (or net vector of the sum) is that last final vector you drew.
Rule 2: Compute Vectors Sums or Differences Using Components
The two mathematical operations of addition and subtraction of vectors are critical to further applications of Newton's Laws of Motion. There are two very simple techniques which lend themselves to a clear physical interpretations that you must master. Recall Newton's second law is expressed as a vector sum: . Because any vector can be written as a set of it components, using the conventional "xyz" cartesian coordinate system, Newton's second law gives rise to a system of three nearly identical scalar equations (the only difference being the identification of the three cartesian directions):
and
and
The three component equations are all straightforward scalar equations. Since all three of these equations are identical in form, we can focus on vectors in a plane (two dimensional) to get the basic operations of vector manipulations graphically. These mathematical procedures are the same whether the situation is 1dimensional, 2D or 3D.
The major steps to vector addition are: first, decomposition (determining the scalar components) and then addition of the components.
 Sum the corresponding components, paying careful attention to the direction (sign) of each component.
 Use the resultant mutually perpendicular component sums to form the resultant vector.
 Use those perpendicular components to draw the resultant vector.
 Use Pythagorean theorem to determine the magnitude of the resultant vector.
 Use appropriate trigonometry to determine the direction of the resultant vector with respect to the arbitrary and convenient coordinate system chosen.
Example: Adding Four Forces in 2D, or a Plane  

All of the essential understanding of vector addition and subtraction can be developed using 2D vectors in a plane. This also makes the "bookkeeping" of terms easier. Therefore choose a coordinate system pointofview that reduces the vectors to a single plane if possible. This will simplify your problem solving. This example shows how to calculate the Net Force when the following four forces (2D) act on a single body.
The formal expression for the Net force should be written down to initiate the analysis that will follow. Include the vector arrows over each force label to distinguish the whole vector from the strength (or magnitude) of that force, F_{1} written using exactly the same labels without the vector arrow.
The net force is determined by following the first four steps mentioned before:

Check Your Understanding
Concept Quiz