Force as a vector is the last part in this course. In SPM, under force as a vector, we need to learn addition of vectors, resolution of vectors, inclined plane and equilibrium of forces.

In this lesson, we will focus on addition of forces, where we will discuss how to use the triangle law and the parallelogram rule to add vectors. After that we will discuss addition of 2 perpendicular forces, where we will learn how to find the magnitude and direction of the resultant vector.

Force is a vector quantity. Addition of forces is same as the addition of vector. If the 2 forces are parallel, we can add or subtract them as the scalar quantities.

However, if the 2 forces are not parallel, we have to use the triangle method or parallelogram rule to complete the task. We must be very sure which method we are using. If we want to use the triangle method, we must join the head of a vector to the tail of another vector and then draw a triangle. The resultant vector must be from the tail of 1 vector to the head of another vector.

If we want to use the parallelogram rule, we must joint the vectors tail to tail or head to head, and then draw a parallelogram. The resultant force is the diagonal of the parallelogram, from tail to head. Again, beware of the direction. Don’t make the direction wrong.

We have just learned that the non-parallel forces can be added by using the triangle method or parallelogram rule, and magnitude and direction can be measure by using ruler and protractor, respectively. However, if the 2 forces are at right angle to each other, we can determine the magnitude and direction easily by using Pythagoras theorem and trigonometry.

For example, we want to add 2 forces, 24N and 7N, which are perpendicular to each other. Let’s use the triangle method. We join the tail of the second vector to the head of another vector. This is the resultant force. The resultant force forms a right angle triangle with the 2 forces. The magnitude is equal to the length of the vector. This can be calculated by using the Pythagoras theorem. The magnitude of the resultant force is equal to the √(24^{2} + 7^{2}) = 25.

The direction of the resultant force can be calculated by using trigonometry. Let’s name this angle *θ*. Tangent *θ* is equal to the opposite side over adjacent side. The length of the opposite side is 7 and the length of the adjacent side is 24. Therefore, *θ* is equal to arc tangent 7/24, which is equal to 16.26^{o}.