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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 specific direction for the sequence of vectors, like clockwise or anti-clockwise, to be maintained. In figure (i), the law is applied starting with vector, b ; whereas the law is applied starting with vector, a , in figure (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 :

$\begin{array}{c}\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}\end{array}$

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

$\begin{array}{c}⇒\mathbf{AB}+\mathbf{BC}=\mathbf{AC}\end{array}$

Adding vector CA on either sides of the equation,

$\begin{array}{c}⇒\mathbf{AB}+\mathbf{BC}+\mathbf{CA}=\mathbf{AC}+\mathbf{CA}\end{array}$

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

$\begin{array}{c}⇒\mathbf{AB}+\mathbf{BC}+\mathbf{CA}=0\end{array}$

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.

Parallelogram law

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

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 figure :

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.

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.

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 ${n}^{\mathrm{th}}$ side, closing the polygon in the opposite direction, represents the sum of the vectors in both magnitude and direction.

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