Vector addition : graphical method

Let us examine the example of displacement of a person in two dierent directions. The two displacement vectors, perpendicular to each other, are added to give the resultant vector. In this case, the closing side of the right triangle represents the sum (i.e. resultant) of individual displacements AB and BC.

Displacement

Figure 1.39

AC = AB + BC (1.1)

The method used to determine the sum in this particular case (in which, the closing side of the triangle represents the sum of the vectors in both magnitude and direction) forms the basic consideration for various rules dedicated to implement vector addition.

1.6.1.1 Triangle law

In most of the situations, we are involved with the addition of two vector quantities. Triangle law of vector addition is appropriate to deal with such situation.

Denition 1.6: Triangle law of vector addition

If two vectors are represented by two sides of a triangle in sequence, then third closing side of the triangle, in the opposite direction of the sequence, represents the sum (or resultant) of the two vectors in both magnitude and direction.

Here, the term sequence means that the vectors are placed such that tail of a vector begins at the arrow head of the vector placed before it.

Triangle law of vector addition

Figure 1.40

The triangle law does not restrict where to start i.e. with which vector to start. Also, it does not put conditions with regard to any specic direction for the sequence of vectors, like clockwise or anti-clockwise, to be maintained. In gure (i), the law is applied starting with vector,b; whereas the law is applied starting with vector, a, in gure (ii). In either case, the resultant vector, c, is same in magnitude and direction.

This is an important result as it conveys that vector addition is commutative in nature i.e. the process of vector addition is independent of the order of addition. This characteristic of vector addition is known as

commutative property of vector addition and is expressed mathematically as :

a + b = b + a (1.2)

If three vectors are represented by three sides of a triangle in sequence, then resultant vector is zero.

In order to prove this, let us consider any two vectors in sequence like AB and BC as shown in the gure.

According to triangle law of vector addition, the resultant vector is represented by the third closing side in the opposite direction. It means that :

Three vectors

Figure 1.41: Three vectors are represented by three sides in sequence.

⇒ AB + BC = AC Adding vector CA on either sides of the equation,

⇒ AB + BC + CA = AC + CA

The right hand side of the equation is vector sum of two equal and opposite vectors, which evaluates to zero. Hence,

Three vectors

Figure 1.42: The resultant of three vectors represented by three sides is zero.

⇒ AB + BC + CA = 0

Note : If the vectors represented by the sides of a triangle are force vectors, then resultant force is zero.

It means that three forces represented by the sides of a triangle in a sequence is a balanced force system.

1.6.1.2 Parallelogram law

Parallelogram law, like triangle law, is applicable to two vectors.

Denition 1.7: Parallelogram law

If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal of parallelogram through the common point represents the sum of the two vectors in both magnitude and direction.

Parallelogram law, as a matter of fact, is an alternate statement of triangle law of vector addition. A graphic representation of the parallelogram law and its interpretation in terms of the triangle is shown in the gure :

Parallelogram law

Figure 1.43

Converting parallelogram sketch to that of triangle law requires shifting vector, b, from the position OB to position AC laterally as shown, while maintaining magnitude and direction.

1.6.1.3 Polygon law

The polygon law is an extension of earlier two laws of vector addition. It is successive application of triangle law to more than two vectors. A pair of vectors (a, b) is added in accordance with triangle law. The intermediate resultant vector (a + b) is then added to third vector (c) again, successively till all vectors to be added have been exhausted.

Successive application of triangle law

Figure 1.44

Denition 1.8: Polygon law

Polygon law of vector addition : If (n-1) numbers of vectors are represented by (n-1) sides of a polygon in sequence, then n^th side, closing the polygon in the opposite direction, represents the sum of the vectors in both magnitude and direction.

In the gure shown below, four vectors namely a, b, c and d are combined to give their sum. Starting with any vector, we add vectors in a manner that the subsequent vector begins at the arrow end of the preceding vector. The illustrations in gures i, iii and iv begin with vectors a, d and c respectively.

Polygon law

Figure 1.45

Matter of fact, polygon formation has great deal of exibility. It may appear that we should elect vectors in increasing or decreasing order of direction (i.e. the angle the vector makes with reference to the direction of the rst vector). But, this is not so. This point is demonstrated in gure (i) and (ii), in which the vectors b and c have simply been exchanged in their positions in the sequence without aecting the end result.

It means that the order of grouping of vectors for addition has no consequence on the result. This characteristic of vector addition is known as associative property of vector addition and is expressed mathematically as :

( a + b ) + c = a + ( b + c ) (1.3)

1.6.1.3.1 Subtraction

Subtraction is considered an addition process with one modication that the second vector (to be subtracted) is rst reversed in direction and is then added to the rst vector. To illustrate the process, let us consider the problem of subtracting vector, b, from , a. Using graphical techniques, we rst reverse the direction of vector, b, and obtain the sum applying triangle or parallelogram law.

Symbolically,

a − b = a + ( − b ) (1.4)

Subtraction

Figure 1.46

Similarly, we can implement subtraction using algebraic method by reversing sign of the vector being subtracted.