The parallelogram law of vector addition provides a geometric method to find the resultant of two vectors originating from the same point. It also provides the mathematical formulas to calculate the magnitude and direction of this resultant vector.
The Parallelogram Law Explained
Imagine two vectors, A and B, starting from the same origin point. According to the parallelogram law, if you place these two vectors as the adjacent sides of a parallelogram, the diagonal of the parallelogram starting from the same origin point represents the resultant vector R.
This resultant vector R is the single vector that produces the same effect as the combined effect of vectors A and B.
Formulas for Resultant Vector
Based on the parallelogram law and trigonometric principles, the magnitude and direction of the resultant vector R can be calculated.
The provided reference states:
The addition of two vector A and vector B is resultant vector R And R = A2 + B2+ 2AB CosΘ1/2And tan β = B SinΘ/ A + B CosΘ Where Θ is the angle between vector A and vector B And β is the angle which vector R makes with the direction of vector A.
Let's break down these formulas:
Magnitude of the Resultant Vector (R)
The formula for the magnitude of the resultant vector R, as given in the reference, is:
- R = A² + B² + 2AB CosΘ¹/²
In standard notation, this formula is typically written using the square root symbol:
- R = √(A² + B² + 2AB CosΘ)
Where:
Ris the magnitude of the resultant vector.Ais the magnitude of vector A.Bis the magnitude of vector B.Θ(Theta) is the angle between vector A and vector B.
Direction of the Resultant Vector (β)
The formula for the direction of the resultant vector R, relative to vector A, as given in the reference, is:
- tan β = (B SinΘ) / (A + B CosΘ)
Where:
β(Beta) is the angle that the resultant vector R makes with the direction of vector A.Ais the magnitude of vector A.Bis the magnitude of vector B.Θ(Theta) is the angle between vector A and vector B.
To find the angle β itself, you would take the inverse tangent (arctan or tan⁻¹) of the value calculated from the right side of the equation.
Applying the Formulas
These formulas are widely used in physics and engineering to solve problems involving forces, velocities, displacements, and other vector quantities.
- Example: To find the single force equivalent to two forces acting on an object, you would use these formulas, treating the forces as vectors A and B.
- Use Case: Calculating the net displacement of an object that moves in two different directions.
Summary Table
| Element | Formula (as per reference) | Standard Formula (Magnitude) | Description |
|---|---|---|---|
| Magnitude (R) | R = A² + B² + 2AB CosΘ¹/² | R = √(A² + B² + 2AB CosΘ) | Length of the resultant vector |
| Direction (β) | tan β = (B SinΘ) / (A + B CosΘ) | tan β = (B SinΘ) / (A + B CosΘ) | Angle of the resultant relative to vector A |
These formulas are fundamental tools derived from the parallelogram law to quantify vector addition.
