To add and subtract the magnitudes of vectors, we must first understand that adding and subtracting vectors involves considering both their magnitude and direction. Here's how it works, drawing from the provided reference:
Understanding Vector Addition and Subtraction
Vector addition and subtraction is not as simple as adding or subtracting their scalar magnitudes. It involves a geometric process, because it involves both the magnitude (how big a vector is) and direction (where it's pointing) of each vector. The provided reference focuses on this process.
Vector Addition
- When adding two vectors, we're essentially finding the resultant vector - a single vector that represents the combined effect of the original two.
- Graphically, we can add vectors by placing the tail of the second vector at the head of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the second.
- Mathematically, we add their component vectors.
- The magnitude of the resultant vector is not simply the sum of the magnitudes of individual vectors (unless the vectors point in the same direction).
Vector Subtraction
- The reference highlights that vector subtraction can be understood as a specific case of vector addition: subtracting a vector is the same as adding its negative.
- To subtract vector B from vector A, written as A - B, you take the negative of B and then add it to A, which is written as A + (-B).
- A negative vector has the same magnitude as the original vector but points in the opposite direction.
Practical Insight
Consider two vectors, A and B, representing two forces acting on an object.
Example of Vector Addition
- A has a magnitude of 5 units and points to the right.
- B has a magnitude of 3 units and points to the right.
- When adding vectors, A + B, because they have the same direction, the magnitude of the resultant vector is 5+3 = 8 units, and the resultant vector also points to the right.
Example of Vector Subtraction
- If, however, B has a magnitude of 3 units and points to the left.
- Then -B has the same magnitude, 3 units, but points to the right.
- To subtract B from A, that's A - B which equals A + (-B).
- The new magnitude becomes 5 + 3 = 8, but since it is A - B, and they are in opposite directions, it is 5-3 = 2 units in the direction of A.
Summary Table
| Operation | Description | Magnitude Change (General Case) | Direction Change (General Case) |
|---|---|---|---|
| A + B | Adding vector B to vector A | The magnitude of the resultant vector is not generally A + B | Changes according to the direction of both vectors |
| A - B | Adding the negative of vector B to vector A | The magnitude of the resultant vector is not generally A - B | Changes according to the direction of both vectors |
Key Takeaways
- Adding vectors involves geometric addition, not simply adding magnitudes.
- Subtracting a vector is the same as adding its negative.
- The negative of a vector has the same magnitude but points in the opposite direction.
- To find the actual magnitude of the resultant vector, you generally need to use vector algebra, such as the law of cosines or the pythagorean theorem for perpendicular vectors.
