Triangle Law of Vector Addition: Definitions, Formula, Derivations, Conditions and Applications

Triangle Law of Vector Addition Overview

The Triangle Law of Vector Addition is a mathematical concept used to calculate the sum of two vectors. This law is used to add two vectors when the first vector's head is connected to the tail of the second vector and then the tail of the first vector is linked to the head of the second vector to form a triangle and thus produce the resultant sum vector. As a result, the triangle law of vector addition is also known as the head-to-tail technique of vector addition.

What is Vector Addition?

One of the vector addition rules is the triangle law of vector addition. Because vectors do not obey ordinary algebraic principles, vector addition is defined as the geometrical sum of two or more vectors. The resulting vector is known as a vector composition. Two laws can be used to do vector addition. They are as follows-

1. Triangle Vector Addition Law
2. Vector Addition Law of Parallelograms

Read more about the Area of Parallelograms.

What is the Triangle Law of Vector Addition?

When two or more vectors are combined, the triangle law of vector addition is applied to find the resultant total vector. Assume we have an automobile traveling from point A to point B. When it reaches point B, it begins traveling again until it reaches point C. To calculate the net displacement of the automobile, we employ the vector addition method. The vector AC gives the net displacement of the automobile, which can be computed using the triangle rule of vector addition-

Read more about the SI Unit of Displacement and the Difference Between Distance and Displacement.

Triangle Law of Vector Addition Formula 

Consider two vectors P and Q with an angle of and the vector R produces their consequent sum vector using the triangle law of vector addition. The formula for the size and direction of the resulting vector R using triangle law for vector addition is,

Triangle Law of Vector Addition Derivation

The triangle law of vector refers to a law that aids in vector algebra and obtaining the consequent of the sum of vectors of two or more vectors. In the example of an automobile, which goes from point A to point B. When the automobile arrives at point B, it will resume its journey to point C (Bastos et al. 2021). In this case, the notion of vector addition may be utilized to calculate a car's net displacement. The net displacement of the automobile is supplied by the AC vector, which may be computed using the triangle rule of vector addition.

In another situation, there are two vectors: Q and P. It is necessary to find the summing of two vectors in order to deduce the vector addition. The Q vector is then shifted such that its direction and magnitude remain unchanged, and its tail connects to the P vector's head. In this situation, the sum of two vectors P and Q may be calculated using the triangular rule of vector addition.

Conditions for the Triangle Law of Vector

For two vectors, the triangle law of vector addition has several requirements. The following are some of the conditions for the triangle law of vector addition-

1. The commutative property is followed by vector addition. i.e   https://d13loartjoc1yn.cloudfront.net/article/1693207691_Commutative Property of Vector Addition .jpg
2. The resultant is 0 if all three sides of the triangle are in the same order.
3. The two vectors to be added must be in the same order. That is, the origin of one vector must correspond with the terminal point of the other. 
4. Vector addition, like addition, has an associative feature, i.e.

Triangle Law of Vector 

The Triangle rule of vectors refers to the phenomena in which two vectors are represented in direction and magnitude by the two neighboring sides of one triangle taken in sequence. Then, in the case of the triangle's closing side, their resultant is calculated in reverse order (Raskin et al. 2021). The Triangle rule of vectors was used to determine the resultant of two vectors: P and Q acting at the angle.

Application of Triangle Law of Vector Addition

One of the most important vector addition rules is the triangle law. Vector addition may be defined as the geometrical sum of two vectors that do not obey conventional algebraic rules (De Sousa et al. 2017). The resulting vector is known as the vector composition. The vector addition is appropriate in numerous situations, which are mentioned below-

1. Scalars and vectors cannot be joined together.
2. The nature of the two vectors that are added must be the same. 
3. For instance, it might be said that velocity must be added to velocity but not to force.

Read more about the Unit Vector Formula.

Conclusion

If the two vectors act in the same direction, the angle between them is determined as follows: = 0°, cos 0° = 1, sin 0° = 0. In this case, the resulting vector will act in the same direction as each vector, and its magnitude total will be equal to that of the other vector. If the two vectors act in opposing directions, the angle between them is determined as follows: = 180°, cos 180° = -1, sin 180° = 0. In this case, the magnitude of the resulting vector is equal to the difference in magnitude of two vectors that act in the direction of the larger of the two vectors.