How to calculate the resultant of two vectors easily?

In this post, we will discuss how to calculate the resultant of two vectors easily. The resultant of two vectors can be done in different methods like (1) Using the Triangle Law, (2) Using the Law of Parallelogram, and (3) using Rectangular Components & Pythagoras Theorem. Among these three methods, the third one is quite handy to solve vector numerical problems.

In fact, this method of using rectangular components of vectors and then finding out the net resultant using the Pythagoras formula is good enough to find out the resultant of more than 2 vectors as well with equal ease.

In our Vector class notes, we have discussed the first two methods in detail. But in this post, we will focus on the smart method of Rectangular Components & Pythagoras theorem to find out the resultant of two vectors easily.

How to use Rectangular Components & Pythagoras theorem to find out the resultant of vectors?

We will use an example in the form of a numerical problem and solve it step by step to understand this method of Rectangular Components.

Rectangular components are nothing but the net horizontal component and the net vertical component of all the vectors being added. We get these by Vector resolution. As you get these two components you have to use the Pythagoras theorem to find out the magnitude of the resultant vector or the vector sum of all the vectors being added.

Problem statement: Calculate the magnitude and direction of the resultant of this combination of vectors.

Solution:

As one of the forces acts only in the horizontal direction, it is straightforward to deal with.

A possible solution here is to take the 40 N vector and split it into horizontal and vertical components. We then combine both the horizontal vectors to give the total horizontal component.

This is combined with the vertical component to give the resultant.

The resultant can be calculated using Pythagoras:

The angle can be calculated using simple trigonometry:

Therefore θ = 8.9°

Thus by using Rectangular Components & Pythagoras theorem, we can easily find out the resultant vector (both magnitude and direction) of two or more vectors easily.

How to calculate the resultant of two vectors easily?
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