Parallelogram Law of Vector Edition and its Derivation

Yashwant Parihar,

When students first start learning vectors in physics, one of the most important and useful concepts they come across is the vector addition parallelogram law. This law helps you understand how two different vectors combine to form a single resultant vector, both visually and mathematically. In this post, we will take a calm, step-by-step approach to state the parallelogram law of vector addition in clear and simple language, so you don’t feel confused by heavy formulas or complex diagrams.

You will also learn how to state and prove the parallelogram law of vector addition using an easy-to-follow derivation that is perfect for school exams, CBSE board preparation, and competitive tests like JEE and NEET. By the end of this lesson, you won’t just memorise the law — you’ll actually understand how and why it works in real physics problems.

What is Parallelogram Law?

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

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

Diagram of Parallelogram of Vector Edition
Diagram of a Parallelogram of Vector Edition

State and Prove the Parallelogram Law of Vector Addition

In Triangle AXD given in the above diagram,
Using Pythagoras Theorem,
AD² = DX² + XA²

R² = (P sin θ)² + (Q + P cos θ)²
R² = P² sin² θ + Q² + P² cos² θ + 2PQcosθ
R² = P² (sin² θ +cos² θ) + Q² + 2PQcosθ
R² = P² + Q² + 2PQcosθ
R = √P² + Q² + 2PQcosθ

Case 1 of Parallelogram Law of Vector Edition

IF θ = 0° than,
R = √P² + Q² + PQcos0°
R = √P² + Q² + 2PQ

Case 2 of the 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 – 2PQ

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