# Parallelogram Law of Vector Edition and its Derivation

Welcome to Thecscience, in this post, we will discuss an important topic of class 11 physics called Parallelogram Law. In this article, we will learn the definition of parallelogram law, its proof, and the law of parallelogram in detail. First, we will talk about the Parallelogram Law of Vector Edition.

## Definition of Parallelogram Law

According to this law, If two vectors represent the adjacent side of a parallelogram then the resultant vector will be diagonal of the parallelogram.

Let the two vectors of a parallelogram P and Q represent the adjacent side of a parallelogram then the R vector will be the resultant vector.

## Derivation of Parallelogram Law of Vector Edition

In Triangle AXD,

Using Pythagoras Theorem,

AD² = DX² + XA²

Note:- If the alpha angle is with the base then cos theta will be written and the perpendicular name as sin theta.

R² = (P sin θ)² + (Q + P cos θ)²

R² = P² sin² θ + Q² + P² cos² θ + PQcosθ

R² = P² (sin² θ +cos² θ) + Q² + PQcosθ

R² = P² + Q² + PQcosθ

R = √P² + Q² + PQcosθ

### Case 1 of Parallelogram Law of Vector Edition

IF θ = 0° than,

R = √P² + Q² + PQcos0°

R = √P² + Q² + PQ

### Case 2 of Parallelogram Law of Vector Edition

IF θ = 90° than,

R = √P² + Q² + PQcos90°

R = √P² + Q²

### Case 3 of Parallelogram Law of Vector Edition

IF θ = 180° than,

R = √P² + Q² + PQcos180°

R = √P² + Q² – PQ

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