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In this article, we shall discuss vector addition as a concept. Vector addition is taught using the triangle law and the parallelogram law, in addition to the commutative and associative features of vector addition.
Triangle law of vector addition
If two vectors are represented by two sides of a triangle in magnitude and direction taken in the same order, then the third side of that triangle represents (in magnitude and direction) the resultant of the vectors.
The Triangle Law of Vector Addition is a mathematical formula that describes how vectors are added together in a triangle.
The displacement from point A to point B is represented by a vector AB, which may be written as considering the following scenario: a boy travels from point A to point B and subsequently from point B to point C. What is the total amount of displacement he has caused from point A to point C?
This displacement is provided by the vector AC, where AC is the acceleration vector.
AC = AB +BC.
In vector addition, this is known as the Triangle Law of Vector Addition.
If you have two vectors a and b, you must place them in such a way that the starting point of one vector coincides with the terminal point of the other vector in order to add them together. This is depicted in Figs. 2 (i) and (ii) below, respectively:
Figure 2 (ii) shows that the vector b has been relocated without affecting its magnitude or direction so that its initial point coincides with the terminal point of vector a. This aids in the formation of the triangle ABC, and the third side, AC, provides us with the sum of the two vectors a and b respectively. As a result, based on Fig. 2 (ii), we can write
AB + BC = AC
After all, we now know that AC = – CA. As a result of the above equation, we have
AB + BC =-CA
Alternatively, AB + BC + CA = 0.
Alternatively, AA = 0.
Additionally, it is a zero vector because the initial and terminal points coincide, as illustrated in the following diagram:
Now, as shown in Fig. 4 below, let’s construct a vector BC′, whose magnitude is the same as that of vector BC, but whose direction is the inverse of that of vector BC. Or,
BC′ =– BC.
Because of this, when we use the Triangle law of vector addition, we obtain
AC′ = AB + BC′ = AB + (– BC′) = a – b.
The difference between vectors a and b is represented by the vector AC′.
Parallelogram law of vector addition
The law of parallelogram vector addition states that if two vectors are represented by two adjacent sides of a parallelogram by direction and magnitude, then the resultant of these vectors is represented (in magnitude and direction) by the diagonal of the parallelogram beginning at the same point as the two original vectors.
The Parallelogram Law of Vector Addition is a mathematical formula that describes how vectors are added together in a parallelogram.
Now, let’s take a look at a slightly more complicated issue. Consider the scenario of a boat travelling down a river from one bank to the other in a direction that is perpendicular to the flow of the river. This boat is subjected to the effects of two velocity vectors:
The boat’s velocity is determined by the engine’s output.
The river’s flow velocity is measured in metres per second.
When these two velocities have an impact on the boat at the same time, the boat begins to move with a different velocity. Examine how we can compute the boat’s resulting velocity and see what we come up with.
To find the solution, consider the two vectors a and b provided below, which are the two neighbouring sides of a parallelogram in terms of their size and direction, as illustrated in the diagram below.
They are represented in size and direction by the diagonal of the parallelogram passing through their common point, which represents their sum, a + b. In vector addition, this is known as the Parallelogram law of vector addition.
Please keep in mind that the Triangle law allows us to deduce the following from Fig. 5.
OA +AC = OC.
Alternatively, OA + OB OC…due to the fact that AC=OB
As a result, we can conclude that the triangle and parallelogram rules of vector addition are similar to one another in terms of functionality.
Image showing the Polygon Law of Vector Addition
It is stated that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant can be represented in magnitude and direction by the closing side of a polygon taken in the opposite order.
Conclusion
If two vectors are represented by two sides of a triangle in magnitude and direction taken in the same order, then the third side of that triangle represents (in magnitude and direction) the resultant of the vectors. The Triangle Law of Vector Addition is a mathematical formula that describes how vectors are added together in a triangle. The law of parallelogram vector addition states that if two vectors are represented by two adjacent sides of a parallelogram by direction and magnitude, then the resultant of these vectors is represented (in magnitude and direction) by the diagonal of the parallelogram beginning at the same point as the two original vectors. The displacement from point A to point B is represented by a vector AB. The Parallelogram Law of Vector Addition is a mathematical formula that describes how vectors are added together in a parallelogram. It is stated that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant can be represented in magnitude and direction by the closing side of a polygon taken in the opposite order.
