Law of Vector Addition

An object with magnitude and direction is called a vector. Furthermore, such a quantity is also represented mathematically or geometrically. A vector consists of initial points and terminal points, represented by arrows. The arrow is shown with a direction equal to that of the quantity and a length equal to the magnitude of that quantity. Vectors do not have positions, even though they have magnitude and direction. A scalar differs from a vector in that it has a magnitude but no direction. For example, acceleration, velocity and displacement are vector quantities, while mass, time and speed are scalars.

Addition of Vectors

Vectors are represented as a combination of direction and magnitude and they are written with an alphabet and an arrow above them to indicate their direction and magnitude combinations. For example, they may be put together using vector addition and the resulting vector can be expressed as follows:

Before we can learn about the characteristics of vector addition, we must first understand the requirements that must be met when vectors are added together. The following are the terms and conditions: Vectors may only be combined if they are of the same type. For example, acceleration should be applied with simple acceleration, not mass, as previously stated.

Law of Vector Addition

Triangle law of vector addition

The addition of vectors may be accomplished by applying the well-known triangle law, which is referred to as the head-to-tail approach. According to this rule, two vectors may be put together by positioning them together so that the head of the first vector meets the tail of the second vector and vice versa. Consequently, by adding the tail of the first vector to the head of the second vector, we may get the sum vector that is produced. The addition of vectors using the triangle method can be with the following steps:

Suppose there are two vectors: p and q

Now, draw a line PQ representing p with P as the tail and Q  as the head. Draw another line ST representing q with S as the tail and T as the head. Now join the line PT with P as the tail and T as the head. The line PT represents the resultant sum of the vectors p and q.

The line PT represents p + q.

The magnitude of p + q is: 

√p2 + q2 + 2pq cos θ

Where,

p= magnitude of the vector

q= magnitude of the vector

θ= angle between p and q.

Parallelogram law of vector addition

We can learn about addition of vectors through the parallelogram method, and we can also try adding vectors graphically. According to it, if two vectors are co-existent at a point and are depicted in magnitude and direction by the two sides of a parallelogram drawn from a point; Their addition comes by the diagonal of the parallelogram which magnitude as well as direction passing through the same point.

According to it, if two vectors, p and q, denote 2 adjacent sides of the parallelogram, both pointing outwards, the diagonal drawn through the junction of the two vectors represent the addition. Its magnitude is shown by the square of the diagonal of the parallelogram, equal to the sum of the square of the adjacent sides.

Important Points to Keep in Mind When Adding Vectors

There are many things that should be kept in mind while studying the addition of vectors, which are as follows:

● Vectors are expressed as a combination of direction and magnitude and they are represented graphically by an arrow.

● As long as we have the components of a vector, we can figure out what the final vector will look like.

● The addition of vectors can be accomplished through the application of the well-known triangle law, which is also known as the head-to-tail method.

Conclusion

It should be kept in mind that vectors are not added algebraically. Note that vectors whose resultant have to be calculated always behave independently. The addition of vectors is finding the resultant of several vectors acting on a body. In addition to vectors, the resulting vector is independent of the order of vectors. You cannot simply add vectors algebraically. Here are some things one needs to consider mind:

• Vectors are not added algebraically but geometrically.

• When calculating the results of a vector, it behaves independently from that vector.

• You can only add vectors that represent the same quantity. For instance, you can add two vectors that represent acceleration. But not when one vector represents acceleration and the other represents force. 

• Vector addition is both associative and commutative in nature. 

• You cannot add vectors and scalars.