Addition of Vectors

Adding two or more vectors together is called vector addition. To get a new vector equal to the total of the previous ones, we add two or more vectors together using the addition operator. The use of vectors to express displacement, velocity, and acceleration means that vector addition can be used to physical quantities. Let’s use the example of a car traveling 10 miles north and south to further comprehend this. Even though the entire distance traveled is 20 miles, there is no displacement here. The vector amounts of the North and South displacements cancel each other out when they are in the opposite directions. 

Vector addition is a mathematical operation:

Vectors are written with an alphabet and an arrow over them (or) with an alphabet written in bold as a mix of direction and magnitude. a + b can be written as the result of combining two vectors, a and b, using vector addition. To understand the properties of vector addition, we must first understand the conditions under which vectors should be added. The following are the conditions:

• Only one type of vector can be inserted at a time. It is preferable to add acceleration, rather than mass, to an object’s acceleration.

• Vectors and scalars can’t be added together.

Consider the following two vectors: D and C. Then, D is equal to Dxi+ Dyj+ Dzk and C is equal to Cxi + Dyj+ Czk. R = C + D is then the resulting vector (or vector sum) and it equals to (Cx + Dx)i + (Cy + Dy)j + (Cz + Cz)k

Graphically Adding Vectors:

Mathematical and graphical approaches are available for adding vectors. The following are the steps involved:

• Adding Vectors Using the Components 

• Triangle Law of Vectors Addition

• Parallelogram Law of Vector Addition

Adding Components to a Vector:

Cartesian vectors can be broken down into their vertical and horizontal axes. An angle vector A can be divided into its vertical and horizontal components as indicated in the below-given image.

There are two components of vector A depicted in this image:

• Ax is the component of vector A along the x-axis, and

• Ay represents the y-axis component of vector A.

Observing the three vectors, we see that they form a right triangle and that A may be written as:

Ax + Ay= A 

Components of a vector can be deduced from its magnitude and angle by using this formula:

A cos Φ= Ax

A sin Φ = Ay

Assuming that the vertical and horizontal components of the two vectors are known, the resulting vector can be computed from this information.

Vector addition’s principles or laws:

Vector addition’s principles:

The addition of vectors is governed by two laws.

• The law of the triangle (triangle law)

• The law of parallelism (Law of Parallelogram)

To demonstrate that the sum of two vectors can be obtained by attaching them head to tail and that the vector sum can be given by the vector that connects the free tail and the free head, we will use the following two laws:

The Triangle Law of Vector Addition:

You can utilize the triangle law, which is also known as the “head to the tail method,” to add vectors. Adding two vectors together according to this law requires that the first vector’s head be attached to the second vector’s tail. Thus, the resultant vector sum may be obtained by connecting the tail of the first vector to the head of the second vector. 

Applying the triangle law, the following steps can be used to add vectors:

• Two vectors M and N are arranged so that the head of M joins the tail of N, and so on.

• Finally, a resultant vector S is constructed that links the tails of M and N to find the sum.

• The resulting, or sum, vector, S, can be stated as S = M + N in the following graphic.

We can see that the original two vectors M and N, along with the sum S, create a triangle when they are added together according to the triangle law.

Law of Parallelograms for Vector Addition:

The parallelogram law of the addition of vectors is another method for combining vectors. p and q are the two vectors we’ll use. Their magnitude and direction form two adjacent sides of a parallelogram. The parallelogram’s diagonal via their common point represents the sum p + q in both magnitude and direction. Vectors can be added using the Parallelogram law. 

Note: The parallelogram law and the triangle law of vector addition are the same things.

Formulas for vector addition:

• Two vectors, a and b, are added together using one of the formulas below

a = <a1, a2, a3> and b = <b1, b2, b3>.

• There are three components, each of which adds up to three additional vectors, hence the sum of these three vectors is equal to the product of their component parts. Here, a + b = <a1 + b1, a2 + b2, a3 + b3>.

• The vector that connects the free heads and free tails is the sum of the two vectors if they are attached at the head and tail (by triangle law).

• When two vectors represent the two neighboring sides of a parallelogram, the sum is a diagonal vector drawn from the intersection of the two vectors (by parallelogram law).

Conclusion:

We have learned about the addition of vectors and their formula.  While you study the addition of vectors, here are some things to keep in mind:

• When representing a vector, we use an arrow to show both the direction and the magnitude of the vector.

• It is possible to calculate the final vector from the components of a vector.

• The head-to-tail approach, which takes advantage of the well-known triangle law, can be used to add vectors.