Addition And Subtraction Of Vectors-Graphical Method

It is important to have an idea about vectors before going forward with a graphical representation of the addition and subtraction of vectors. It is interesting to note that you cannot use the same method to add two vectors together as simply as you do with numbers. Remember that a vector includes both directions as well as magnitude. A vector does not have any fixed position, however. It remains unaltered if displaced in a direction that is parallel to the original position.

You have to represent a set of vectors graphically before proceeding to add them or subtract one from the other. You are sure to understand the concept better by considering the displacement when a vehicle travels from point A to point B. Traveling from 10 miles to the North of a point to 10 miles South of it is not calculated as 20 miles though. Sure, the total distance traveled is 20 miles but you have to take its displacement into account as well. The North and South are directions that cause some displacement. But traveling 10 miles in both directions will cancel out the total displacement bringing it back to 0. 

The procedure for addition and subtraction of vectors are a little different as compared to scalars too. You can always define the laws involved in addition and subtraction of vectors in the following manner. Here are the details. Do check them out…

Addition of vectors

The very first aspect that you would have to consider about adding two vectors together is to remember a few rules related to the addition and subtraction of vectors. The following tips will enable you to do it graphically as well.

• You have to use geometry to add two vectors and not algebra
• The calculation may be about finding the sum of each vector but the vectors behave independently and follow specific laws individually
• The final result equals the count of vectors acting on a particular body at any given point in time
• The result of vector addition is cumulative. In other words, the resultant number/direction is not dependent on the order of the vectors that are being calculated

You may illustrate the process of summation of two or multiple vectors by providing vector addition and subtraction examples such as:-

Some of the other laws to consider before attempting addition and subtraction of vectors are:-

Triangle Law

Adding two vectors represented by a  and b can be done by drawing a line AB where A forms the tail and B is the head.

• Draw a second line BC where B forms the tail and C its head
• Join the points A and C by another line. The resulting line AC will give you the sum of the two vectors

Parallelogram Law

The addition and subtraction of vectors, particularly the procedure of addition, can be explained with a parallelogram too. The law elaborates the process in the following manner:-

• P and Q represent two sides of a parallelogram that are adjacent to each other. Each side points in an external direction
• A diagonal line drawn from the intersecting point of P and Q will give the result

Polygon Law

This particular law considers multiple counts of vectors. When their magnitude along with the direction of each is considered in an order of the sides, the result is evident by the magnitude and the direction considered in the opposite direction of its sides. The addition and Subtraction of vectors by using a polygon can be depicted by the following method:-

Name the four vectors as A, B, C, and D consecutively. You now need to obtain the results by adding all four-vectors. You may want to segregate the sides A and B and form a triangle. The opposite direction of both vectors will provide you with the result of adding these two vectors. Now consider the opposite triangle created out of the polygon in the same manner. You will be able to figure out the total of A through D by tracing the direction and taking the total displacement into account.

Subtraction of vectors

The procedure for the addition and subtraction of vectors is not too complex. You will be able to grasp the facts and understand the underlying concept by learning the additional procedure carefully. Go ahead and check the basics of algebra. You know that you can represent the a-b as an addition too. In other words, all you have to do is write it as . You will get an identical answer every time. Apply the conventional formula and arrive at the result by using .

Remember that -b is nothing b in the reverse direction. Yes! You have to learn the concepts related to vectors to understand the laws of nature and calculate all kinds of Physics problems. Being aware of how to do addition and subtraction of vectors will give you a head start.

You may be surprised to learn that while the addition of vectors has numerous uses, the same is not true of subtraction. However, you have to learn it alongside understanding the complexities of Physics.  Try to recall the parallelogram law when you are trying to get a hang of the addition and subtraction of vectors. Draw a parallelogram by using two distinct vectors u and v. You will get the result of addition by drawing a diagonal from the coinciding point of the two vectors. Similarly drawing a diagonal in the opposite direction will give you the result of subtraction.

Vector subtraction is also possible when you have two vectors as components or provided as columns. Thus you may have A=(ax, ay) and B=(bx, by). The result R (difference in vectors) will be depicted as R=A-B. You may also get both the vertical and horizontal representations as Rx = ax – bx and Ry = ay – by respectively. The results related to the difference between the two sets of vectors can be calculated rather easily by computing the vertical and horizontal components.

The head to tail law about addition and subtraction of vectors can be applied perfectly to only subtraction too. A mere reversal of direction for one of the vectors will give you the result without too much trouble. The tail of a vector would coincide with the head of another thus giving you a clear idea of how to find the difference between two vectors.

Conclusion

The graphical method of addition and subtraction of vectors includes drawing the vectors on a graph and then calculating the required result by using the head to tail procedure. The addition of vectors A and B will result in R that is a sum of the two. The vectors are not simple scalars, however. Instead, you have to consider both its magnitude as well as the direction. Taking note of the opposite direction is the most effective method of subtracting vectors. B will thus be represented by –B. Addition and subtraction of vectors are useful for their practical applications and academic interest.