Parallelogram Law – Law of Addition, Law of vectors, Procedure and Proofs

# Parallelogram Law – Law of Addition, Law of vectors, Procedure and Proofs

## What is Parallelogram?

Parallelogram Law – Law of Addition: A parallelogram is a quadrilateral with two pairs of parallel sides.

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If two vectors are placed at right angles to each other, the resultant vector is the sum of the vectors. The magnitude of the resultant vector is the sum of the magnitudes of the vectors and the direction is the direction of the vector sum.

## Parallelogram Law of Vectors

A parallelogram law of vectors states that the vector sum of two vectors is the diagonal of the parallelogram formed by them. The law is also known as the parallelogram law of addition.

## Parallelogram Law of Addition of Vectors Procedure

• Draw a diagram of the two vectors to be added. The vectors should be represented by two arrows, one for each vector.
• Label the start point of the first vector and the end point of the second vector.
• Draw a parallelogram using the two vectors as the sides. The vectors should be perpendicular to each other.
• Find the length of each vector and the angle between the vectors.
• Add the vectors together by finding the vector sum. This is the vector that points from the start point of the first vector to the end point of the second vector.
• Draw the vector sum as an arrow.

## Parallelogram Law Proof

A parallelogram is a quadrilateral with two pairs of parallel sides. The following theorem known as the parallelogram law.

Theorem: The sum of the two opposite angles of a parallelogram is 180°.

Proof:

• We will use the following definition:
• The angle between two lines is the angle between the lines when they considered to be rays.
• Let $\angle A$ and $\angle B$ be the angles opposite the respective sides of length $a$ and $b$.
• Since the two angles are opposite, they are supplementary. That is,
• $\angle A + \angle B = 180^{\circ}$
• Now, we use the fact that the angles of a parallelogram are right angles. Hence,
• $\angle A = 90^{\circ}$ and $\angle B = 90^{\circ}$
• Therefore,
• $\angle A + \angle B = 180^{\circ}$
• $\angle A + \angle B = 180^{\circ}$

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