How to add vectors

Right Triangle Figure 1

How to add vectors? We know that vectors are physical quantities that consist of a magnitude as well as a direction, for example velocity, acceleration, and displacement, as opposed to scalars, which consist of magnitude only, for example speed, distance, or energy. While scalars can be added by adding their magnitudes , vectors are  more complicated to add. When only two vectors are given, first we have to connect those vectors, for example, 4 metre west and 3 metre north. We can choose to draw them on a piece of paper and connect them together to form a right triangle when a hypothetical hypotenuse is added. The second step is to use the Pythagorean Theorem to calculate the length of the hypotenuse.

Pythagorean Theorem Figure 2

And as we calculated, the length is 5 metre. After this step, the final step is to determine the direction and angle to the y-axis. For example, in figure 1, we draw a cross at the starting point, and calculate the angle between the blue line and the y-axis. Figure 3 shows an example. the angle should be calculated is the angle between the y-axis and line A.

Use the equation in Figure 5 and use Figure 4 to determine a right triangle.

Figure 5

Figure 3

Figure 4

However, the above information are just the simplest part of adding vectors. When more and more vectors are given, how to add them all together and find the hypotenuse and angle? In such question, when 3 vectors are given and they are 5 metre 30 degree west north, 3 metre 60 degree east north and 10 metre 20 degree west south, how to solve it? This time a more complicated method would be introduced. We will add them all together, but before that we have to classify the y and x for those hypotenuses given. In figure 3, the vertical dotted line is call the y of a hypotenuse, and the horizontal dotted line is called the x of a hypotenuse. When they are all together, a right triangle could be formed. When the length of a hypotenuse and the angle are given, we use the equations introduced in the below diagram to calculate the adjacent and opposite lines.

Figure 6

When we have found all of the x and y, we will add then together and get a final x and y. We then plot them on a piece of graph paper. The shape formed by plotting the x and y can simply be the one showed in Figure 1. Since x and y are the opposite and adjacent line of the triangle, we can easily find the hypotenuse by using figure 5's equation. Lastly, we repeat the same step for finding the angle and write the answer like 20m (N 30 degree S)