Addition of Vectors

In vector theory, determining the sum of two vectors can be accomplished using a method that is based on the parallelogram law of vector addition. This method is known as the parallelogram law of vector addition. For the purpose of adding two vectors, this law of vector addition is applied when the vectors to be added form two adjacent sides of a parallelogram. The parallelogram is produced by combining the tails of the two vectors to produce the parallelogram itself. In this particular instance, the calculation of the sum of the two vectors makes use of the diagonal of the parallelogram. 

The Triangle Law of Addition to Vectors

Let’s say we have a car that is travelling from point A to point B, as depicted in the diagram on the right. Once it reaches point B, it resumes its journey until it reaches point C. Now, in order to determine the total distance that the car has travelled, we will apply the concept of adding vectors together. The vector AC, which can be calculated by applying the triangle rule of vector addition as demonstrated in the following example, represents the car’s total displacement, which can be expressed as a net value.

AC = AB + BC

In a similar manner, if we have two vectors P and Q, as shown below, and we need to compute the sum of those vectors, we can move the vector Q in such a way that its tail is attached to the head of the vector P without changing the magnitude or direction of the vector Q. This will allow us to find the sum of the two vectors. The following is how we can compute the sum of the vectors P and Q using the triangle law of vector addition:

R = P + Q

Parallelogram Law of Vector Addition

The parallelogram law of vector addition is a law that makes it easier to determine the sum vector that results from an operation known as vector addition, which can be used to determine the sum of two vectors. Assume that a fish is swimming from one side of a river to the other side of the river in the direction of the vector Q, and that the water in the river is flowing in the direction of the vector P, as shown in the image below. This would indicate that the fish is swimming in the direction of the vector Q.

Because there are two velocities at play here, namely the velocity of the fish and the flow of water in the river (which will be a separate velocity), the total of these two velocities is what is meant to be understood as the fish’s “net” velocity. Because of this, the fish will move in a different direction along a vector that has the same length as the sum of the two velocities it is experiencing. It is now possible to calculate the net velocity by thinking of these two vectors as the adjacent sides of a parallelogram and using the parallelogram law of vector addition to determine the total vector that results from adding up all of the vectors in the parallelogram.

Conclusion

Vectors are combinations of direction and magnitude. Using the technique known as vector addition, it is possible to combine two vectors, which are represented by the symbol “a+b,” and the final vector can be expressed as a + b. Only vectors that are of the same type as one another can be combined into a single vector. As an illustration, acceleration ought to be added with just acceleration and not mass as well. Combining vectors and scalars is not something that can be done. If you order the two vectors by attaching the head of one vector to the tail of the other vector, then the vector that sums the two vectors is the vector that unites the free heads and tails of the vectors (by triangle law). If the two vectors are meant to stand in for the two adjacent sides of a parallelogram, then the sum will point to the diagonal vector that can be drawn from the point at which the two vectors intersect, which is also known as the point where the two vectors meet (by parallelogram law).