Subtracting is not as complicated as it sounds, especially if you already know how to add vectors. Like subtracting integers, subtracting vectors is just like adding the opposite of the second vector.
Subtracting Vectors
\(\mathsf{ \overrightarrow{A} - \overrightarrow{B} = \overrightarrow{A} + -\overrightarrow{B} }\)
In other words, you take the opposite DIRECTION for the second vector and add it to the first. After you find the change in direction of the second vector, you use exactly the same techniques you do in adding vectors. When direction is given as left or right, up or down, North or South, or East or West it is easy to find the opposite direction. If the direction is given as an angle measurement, you either have to add or subtract 180° to find the opposite direction. Let's look at a few examples.
| Original Direction | Operation | Opposite Direction |
|---|---|---|
| 25° North of East | 25° + 180° = 205° | 25° South of West |
| 65° East of North | 65° + 180° = 245° | 65° West of South |
| 25° West of North | 115° + 180° = 295° | 25° East of South |
| 45° South of East | 315° - 180° = 135° (subtract if the sum pushes you over 360°) |
45° North of West |
| 32° West of South | 238° - 180° = 58° | 32° East of North |
Question
Without adding or subtracting the 180°, do you notice a short cut?
The angle measures in each case stayed the same, but both the first and second directions flip-flopped. North always changes to South and vice versa. East always changes to West and vice versa.
