The resultant of coplanar vectors using the graphical method is found by drawing the vectors to scale and connecting them appropriately, with the final resulting vector being drawn from the starting point to the ending point of the vector chain.
When dealing with just two coplanar vectors, a common graphical technique is the Parallelogram Method.
The Parallelogram Method (For Two Vectors)
This method is particularly intuitive for visualizing the combined effect of two forces or vectors acting at the same point.
- Draw the Vectors: Choose a suitable scale (e.g., 1 cm = 1 Newton) and draw the two vectors originating from the same point. The length of each vector should represent its magnitude according to the scale, and the direction should be accurate using a protractor.
- Form the Parallelogram: From the tip (head) of the first vector, draw a line parallel to the second vector. From the tip of the second vector, draw a line parallel to the first vector. These parallel lines should intersect, forming a parallelogram.
- Draw the Resultant: The resultant can be represented graphically by the diagonal of the parallelogram formed by using the two force vectors to determine the length of the sides of the parallelogram. This diagonal starts from the common origin point where the two original vectors began and extends to the opposite corner of the parallelogram.
- Measure the Resultant: The magnitude of the resultant can be accurately measured as the scaled length of the diagonal. The direction of the resultant is given by the angle the diagonal makes with a reference axis (like the horizontal).
- Practical Insight: The parallelogram method visually shows how two forces "pulling" in different directions combine into a single effective force.
The Head-to-Tail Method (For Two or More Vectors)
This method is more general and can be used to find the resultant of any number of coplanar vectors.
- Draw the First Vector: Choose a scale and draw the first vector starting from a convenient origin point.
- Draw Subsequent Vectors: Draw the second vector starting from the tip (head) of the first vector. Then, draw the third vector starting from the tip of the second, and so on. Maintain the correct length and direction for each vector according to the scale and given angles.
- Draw the Resultant: The resultant vector is drawn from the origin of the first vector to the tip of the last vector drawn.
- Measure the Resultant: Measure the length of the resultant vector and multiply it by the scale factor to find its magnitude. Measure the angle the resultant makes with a reference axis to find its direction.
- Example: If you have three vectors A, B, and C, you draw A, then draw B starting where A ends, then draw C starting where B ends. The resultant is the vector from the start of A to the end of C.
Summary of Graphical Methods
Here's a quick comparison:
| Method | Number of Vectors | Process | Resultant | Key Benefit |
|---|---|---|---|---|
| Parallelogram | Two | Vectors drawn from origin, form parallelogram | Diagonal from origin to opposite corner | Visualizes vector combination |
| Head-to-Tail | Two or more | Vectors drawn end-to-end in sequence | Vector from start of first to end of last | General method for any number |
Using these graphical methods provides a visual understanding of vector addition and allows for an approximate determination of the resultant's magnitude and direction by careful drawing and measurement.
