Cases for Parallelogram Law of Vector Addition

There are some cases where the Parallelogram Law of Vector Addition can be applied: When

Vectors are Parallel (θ = 0°)

When two vectors are parallel and have the same direction, the law simplifies to simple addition then the resultant vector will have the same direction as the original vectors and a magnitude equal to the sum of their magnitudes.

|R| = √(P²+ Q²+ 2PQcosθ)

Since, θ = 0°

⇒ |R| = √(P² + Q²+ 2PQcos0)

⇒ |R| = √(P² + Q²+ 2PQ)

⇒ |R| = √(P + Q)²

⇒ |R| = P + Q

When two vectors are in the same direction, the resultant vector is their algebraic sum given by using R = P + Q where R is the resultant vector, P and Q are the parallel vectors.

Vectors are in Opposite Direction (θ = 180°)

When two vectors are in opposite directions, the resultant vector will have a magnitude of zero because the vectors cancel each other out.

|R| = √(P²+ Q²+ 2PQcosθ)

Since, θ = 180°

|R| = √(P²+ Q²+ 2PQcos180)

⇒ |R| = √(P²+ Q²– 2PQ)

⇒ |R| = √(P – Q)²

⇒ |R| = P – Q

By using parallelogram law of vector addition R = P – Q, where R is the resultant vector, P is the first vector, and Q is the second vector. When the vectors are in opposite directions, the value of θ = 180 degree.

Vectors are Perpendicular (θ = 90°)

When two vectors are perpendicular to each other, by using the Pythagorean theorem, we can find magnitude of the resultant vector is equal to the square root of the sum of the squares of the magnitudes of the two vectors.

|R| = √(P²+ Q²+ 2PQcosθ)

Since, θ = 90°

⇒ |R| = √(P²+ Q²+ 2PQcos90)

⇒ |R| = √(P²+ Q²– 0)

⇒ |R| = √(P² + Q²)

When vectors are perpendicular, their angles (θ) between them are typically 90 degrees. Using the Parallelogram Law of Vector Addition R = √(P² + Q² + 2PQcosθ) since (θ = 90 degree) so, the formula is R = √(P² + Q²).

Parallelogram Law of Vector Addition explains that when two vectors are considered to be the two adjacent sides of a parallelogram with their tails meeting at the common point, then the diagonal of the parallelogram originating from the common point will be the resultant vector. It is also known as Parallelogram Law in Vector Algebra.

Parallelogram Law of Vector Addition is basically the mathematical expression for vector addition. This law is used to add two vectors when the vectors form two adjacent sides of the parallelogram formed by combining the tails of these two vectors to produce or make the parallelogram itself then the diagonal of the parallelogram is used to calculate the sum of the two vectors which is called resultant vector.

Here, in this article we will learn in detail, the Parallelogram Law of Vector Addition along with a brief introduction to vector addition. We will also learn Parallelogram Law of Vector Addition Formula, Derivation of Parallelogram Law of Vector Addition Formula, its different cases, and its application.