To graph a resultant vector using the triangle method, you visualize the vectors as consecutive displacements. The resultant vector represents the single displacement that takes you from the starting point of the first vector to the ending point of the last vector.
Here's a straightforward guide to applying the triangle method for adding two vectors, say V and W:
Steps for Graphing Vector Sum (V + W)
The triangle method (also known as the head-to-tail method for two vectors) involves placing the vectors end-to-end to form two sides of a triangle. The third side represents the resultant vector.
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Graph the First Vector: Begin by graphing the first vector (V). As mentioned in the reference, it's common practice to graph the initial vector with its initial point at the origin (0,0) of a coordinate system. Draw an arrow from the origin to the point corresponding to the vector's components.
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Place the Second Vector: Now, take the second vector (W) and place its initial point (tail) at the terminal point (head) of the first vector (V). Ensure the second vector maintains its original magnitude and direction; you are essentially sliding it without rotating or changing its length.
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Draw the Resultant Vector: The resultant vector (R = V + W) is then drawn from the initial point of the first vector (which was typically the origin) to the terminal point of the second vector. This resultant vector completes the triangle. The length and direction of this resultant vector represent the magnitude and direction of the sum of the two vectors.
These three vectors—V, W, and R—form the sides of a triangle, illustrating the vector addition V + W = R.
Visualizing the Process
Imagine walking. If you first walk along the path represented by vector V, and then from that ending point, you walk along the path represented by vector W, your final position relative to your starting point is given by the single path represented by the resultant vector R.
| Step | Action | Outcome |
|---|---|---|
| 1. Graph First Vector | Draw V starting at the origin. | V is positioned. |
| 2. Place Second Vector | Place W so its tail is at the head of V. | V and W are head-to-tail. |
| 3. Draw Resultant Vector | Draw R from the tail of V (origin) to the head of W, forming the third side of the triangle. | R = V + W is graphed. |
The length of the resultant vector R on the graph, when measured against the scale used for the original vectors, gives its magnitude. The angle it makes with a reference axis (like the positive x-axis) gives its direction. This graphical method provides a visual representation of vector addition.
