Vector Calculator Parallelogram Law

Parallelogram Law Formula:

\[ R = \sqrt{A^2 + B^2 + 2AB \cos\theta} \]

Resultant (R):

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1. What is the Parallelogram Law of Vector Addition?

The Parallelogram Law is a fundamental principle in vector mathematics that states: "If two vectors are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the parallelogram passing through their common point."

2. How Does the Calculator Work?

The calculator uses the parallelogram law formula:

\[ R = \sqrt{A^2 + B^2 + 2AB \cos\theta} \]

Where:

\( A \) — Magnitude of first vector
\( B \) — Magnitude of second vector
\( \theta \) — Angle between vectors (in degrees)
\( R \) — Resultant vector magnitude

Explanation: The formula calculates the magnitude of the resultant vector when two vectors are added using the parallelogram method, considering both their magnitudes and the angle between them.

3. Importance of Vector Addition

Details: Vector addition using the parallelogram law is crucial in physics and engineering for calculating net forces, velocities, displacements, and other vector quantities in two-dimensional systems.

4. Using the Calculator

Tips: Enter the magnitudes of both vectors (must be positive values) and the angle between them (0° to 180°). The calculator will compute the magnitude of the resultant vector.

5. Frequently Asked Questions (FAQ)

Q1: What happens when θ = 0°?
A: When vectors are parallel (θ = 0°), the resultant is simply the sum of the two magnitudes (R = A + B).

Q2: What happens when θ = 90°?
A: When vectors are perpendicular (θ = 90°), the formula simplifies to the Pythagorean theorem (R = √(A² + B²)).

Q3: What happens when θ = 180°?
A: When vectors are opposite (θ = 180°), the resultant is the difference between the magnitudes (R = |A - B|).

Q4: Can this calculator handle negative magnitudes?
A: No, vector magnitudes are always positive quantities. The direction is accounted for by the angle between vectors.

Q5: What are typical applications of this law?
A: Used in physics for force calculations, engineering for structural analysis, navigation for vector addition of velocities, and computer graphics for vector operations.