In the parallelogram law of vector addition, the diagonal of the completed parallelogram represents the resultant vector in both magnitude and direction.
Understanding the Parallelogram Law
The parallelogram law is a graphical method used to find the sum (or resultant) of two vectors. When adding two vectors using this method, you treat them as adjacent sides of a parallelogram originating from the same point.
According to the Parallelogram Law of Addition of Vectors Procedure:
Treat these vectors as the adjacent sides and complete the parallelogram. Now, the diagonal represents the resultant vector in both magnitude and direction.
How it Works
- Place the vectors: Position the two vectors so their tails meet at a single point.
- Complete the parallelogram: Draw lines parallel to each vector from the head of the other vector. These lines form the remaining two sides, completing the parallelogram.
- Draw the diagonal: Draw a diagonal line from the common tail of the original two vectors to the opposite vertex of the parallelogram.
This specific diagonal is the vector sum. Its length corresponds to the magnitude of the resultant vector, and its direction corresponds to the direction of the resultant vector.
Why the Diagonal Represents the Resultant
The parallelogram law is essentially a visual representation of placing vectors head-to-tail (the triangle law).
Imagine you have vectors A and B starting from the same point.
- If you move vector B so its tail is at the head of A, the vector from the tail of A to the head of the new B is the resultant (A + B).
- Similarly, if you move vector A so its tail is at the head of B, the vector from the tail of B to the head of the new A is the resultant (B + A).
In the parallelogram, the diagonal drawn from the common origin is the same vector obtained by either placing B at the head of A or A at the head of B. It represents the single vector that has the same effect as the combined effect of the original two vectors.
Practical Examples
The parallelogram law is useful in various physics and engineering applications.
- Forces: If two forces are acting on an object at an angle, their combined effect (the resultant force) can be found using the parallelogram law. The diagonal shows the direction and strength of the net force.
- Velocities: If an object is moving with a certain velocity relative to a medium (like a boat in water or a plane in air) and the medium itself is moving, the object's velocity relative to the ground is the resultant of the two velocities.
| Original Vectors | Represented As... | Diagonal Represents... |
|---|---|---|
| Adjacent sides of a parallelogram | Vectors to be added | Resultant vector (Vector Sum) |
Key Takeaway
When using the parallelogram law for vector addition, the diagonal drawn from the point where the two vectors originate is the graphical representation of the resultant vector, embodying both its overall strength (magnitude) and its direction.
