The Parallelogram Law of Vector Addition is a fundamental principle used to find the resultant vector (sum) of two vectors acting at the same point. It provides a geometric method to determine both the magnitude and direction of the resultant.
Understanding the Parallelogram Law
According to the Parallelogram Law of Vector Addition, as stated in the reference: "if two vectors acting simultaneously at a point are represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from that point, then the resultant vector can be represented both in magnitude and direction by the diagonal of the parallelogram that passes through that same point."
Imagine two vectors, A and B, originating from the same point.
- Represent vector A as one side of a parallelogram.
- Represent vector B as an adjacent side of the same parallelogram, starting from the same point as A.
- Complete the parallelogram by drawing lines parallel to A and B from the endpoints of B and A, respectively.
- The resultant vector, R = A + B, is the diagonal of the parallelogram that starts from the same initial point as vectors A and B.
Calculating Magnitude and Direction
While the geometric method gives a visual representation, calculating the exact magnitude and direction of the resultant vector R requires formulas based on the law of cosines and the law of sines.
Let:
- |A| be the magnitude of vector A.
- |B| be the magnitude of vector B.
- θ be the angle between vectors A and B.
- |R| be the magnitude of the resultant vector R.
- α be the angle between the resultant vector R and vector A.
Magnitude of the Resultant Vector (|R|)
The magnitude of the resultant vector R is given by the formula derived from the law of cosines applied to the triangle formed by vectors A, B, and R:
|R| = √(|A|² + |B|² + 2|A||B| cos θ)
- This formula takes into account the angle between the two vectors.
- If the vectors are parallel and in the same direction (θ = 0°), cos θ = 1, and |R| = √(|A|² + |B|² + 2|A||B|) = √((|A| + |B|)² ) = |A| + |B|.
- If the vectors are perpendicular (θ = 90°), cos θ = 0, and |R| = √(|A|² + |B|²), which is the Pythagorean theorem.
- If the vectors are parallel and in opposite directions (θ = 180°), cos θ = -1, and |R| = √(|A|² + |B|² - 2|A||B|) = √((|A| - |B|)² ) = ||A| - |B|| (the absolute difference).
Direction of the Resultant Vector (α)
The direction of the resultant vector R is typically described by the angle it makes with one of the original vectors (e.g., vector A). This angle (α) can be found using the Law of Sines:
|R| / sin θ = |B| / sin α = |A| / sin β (where β is the angle between R and B)
From this, we can solve for sin α:
sin α = (|B| sin θ) / |R|
Alternatively, the angle α can be found using the tangent rule:
tan α = (|B| sin θ) / (|A| + |B| cos θ)
Once tan α is calculated, the angle α can be found using the arctan (inverse tangent) function:
α = arctan[ (|B| sin θ) / (|A| + |B| cos θ) ]
Summary of Formulas
Here's a quick look at the key formulas:
| Property | Formula | Notes |
|---|---|---|
| Magnitude | R | |
| Direction | tan α = ( | B |
Practical Applications
The parallelogram law is widely used in physics and engineering to solve problems involving the combination of vector quantities such as:
- Forces acting on an object.
- Velocities or displacements.
- Electric or magnetic fields.
By applying the parallelogram law, engineers and physicists can predict the net effect (resultant) of multiple vectors acting simultaneously.
