How Do You Subtract Vectors Geometrically?

You subtract vectors geometrically using one of two primary methods: by making them coinitial and drawing a vector between their tips, or by adding the negative of the subtrahend vector.

Vector subtraction can be visualized in two main ways, both yielding the same result for a - b. These methods are fundamental in understanding vector operations without relying on coordinates.

Method 1: The Coinitial Method (Tip-to-Tip)

This method involves placing the vectors so they start at the same point.

1. Make Vectors Coinitial: Position vectors a and b so that their tails (starting points) coincide.
2. Draw the Resultant Vector: Draw a new vector starting from the tip (head) of vector b and ending at the tip (head) of vector a.
3. Identify the Result: This new vector represents a - b.

• Why this works: If you were to add the vector from b's tip to a's tip to vector b, you would end up at the tip of a. This demonstrates that b + (a - b) = a, which is the definition of subtraction.

This approach is explicitly described in the provided reference: "To subtract two vectors a and b graphically (i.e., to find a - b), just make them coinitial first and then draw a vector from the tip of b to the tip of a."

Method 2: Adding the Negative Vector

This method converts the subtraction problem into an addition problem by using the concept of a negative vector.

1. Find the Negative Vector: Determine the vector -b. This is a vector with the same magnitude as b but pointing in the exact opposite direction. You get -b by multiplying vector b by -1.
2. Perform Vector Addition: Now, instead of calculating a - b, you calculate a + (-b). You can do this using standard geometric vector addition methods, such as:
   • Triangle Method: Place the tail of -b at the tip of a. The resultant vector a + (-b) is drawn from the tail of a to the tip of -b.
   • Parallelogram Method: Place the tails of a and -b at the same point. Complete the parallelogram using these two vectors as adjacent sides. The diagonal drawn from the common tail is the resultant vector a + (-b).
3. Identify the Result: The resultant vector from the addition is a - b.

The reference also highlights this method: "We can add -b (the negative of vector b which is obtained by multiplying b with -1) to a to perform the vector subtraction a - b."

Both methods are valid and provide the same geometric result for vector subtraction. The first method often feels more direct when visualizing the difference between two vectors starting from the same point, while the second leverages the established rules of vector addition.