The direction of the resultant vector in the parallelogram law of vector addition is along the diagonal of the parallelogram that originates from the common point of contact of the two vectors being added.
Understanding the Parallelogram Law
Vectors are fundamental quantities in physics and engineering that possess both magnitude (size) and direction. When adding two vectors, their combined effect is represented by a single vector called the resultant vector. The parallelogram law is a graphical method for finding this resultant vector.
How the Law Works
According to the Parallelogram law of vector addition, if two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram drawn from a common point, then their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram from the point of contact of the two vectors.
Let's break this down:
- Represent the Vectors: Draw the two vectors, say vector A and vector B, from the same starting point (origin).
- Form the Parallelogram: Use vector A and vector B as two adjacent sides of a parallelogram. Complete the parallelogram by drawing lines parallel to A and B from the heads of B and A, respectively, until they meet.
- Identify the Resultant: The resultant vector, R, is the diagonal that starts from the origin (the common point of contact of A and B) and extends to the opposite vertex of the parallelogram.
The Direction of the Resultant
Based directly on the parallelogram law, the direction of the resultant vector R is specifically:
- Along the diagonal of the parallelogram.
- Starting from the point of contact where the two original vectors (A and B) originate.
This means the resultant vector points from the tail of the original vectors towards the opposite corner of the completed parallelogram.
Visualizing the Direction
Imagine two forces acting on an object at the same point. If you represent these forces as vectors forming two sides of a parallelogram, their combined effect (the resultant force) acts in the direction of the diagonal starting from that point.
| Vector Operation | Geometric Shape | Resultant Direction |
|---|---|---|
| Parallelogram Vector Add. | Parallelogram | Along the diagonal from the origin |
Understanding this direction is crucial for analyzing the combined effect of multiple forces, velocities, or other vector quantities acting simultaneously. The magnitude of the resultant can be calculated using trigonometry (like the Law of Cosines), but its direction is visually represented by that specific diagonal.
