To subtract vectors in linear algebra, you essentially add the first vector to the negative of the second vector.
Subtracting vector V from vector U, written as U - V, is performed by adding U and the negative of V, denoted as U + (-V). This key concept allows you to leverage the rules and methods you use for vector addition.
Vector subtraction builds directly upon the concepts of vector addition and scalar multiplication (specifically, multiplication by -1). When you perform U - V:
- You find the negative of vector V.
- You add vector U to this negative vector, -V.
What is the Negative of a Vector?
The negative of a vector V, written as -V, is a vector that has the exact same magnitude (length) as V but points in the opposite direction. As stated in the referenced material, -V is simply V but pointing in the opposite direction.
Methods for Subtracting Vectors
There are two primary ways to conceptualize and perform vector subtraction: graphically and algebraically.
1. Graphical Vector Subtraction (Head-to-Tail Method)
This method visualizes the subtraction as an addition problem: U + (-V).
- Step 1: Draw vector U.
- Step 2: Draw the negative of vector V, which is -V. Remember, -V is parallel to V but points the other way.
- Step 3: Place the tail of the negative vector (-V) at the head (tip) of vector U. This is the head-to-tail method for addition, as mentioned in the reference ("we're really just doing U. Plus negative V so we're just going to take the head to tail.").
- Step 4: The resulting vector, U - V, is drawn from the tail of U to the head of -V.
This resulting vector visually represents the difference between U and V.
Example (Graphical):
Suppose you have vector U pointing right and slightly up, and vector V pointing right and slightly down.
- Draw U.
- Draw -V (pointing left and slightly up).
- Place the tail of -V at the head of U.
- Draw the vector from the tail of U to the head of -V. This is U - V.
2. Algebraic Vector Subtraction (Component-wise)
If vectors are given in terms of their components, subtraction is straightforward. You simply subtract the corresponding components of the second vector from the first vector.
Let U = <u₁, u₂, ..., uₙ> and V = <v₁, v₂, ..., vₙ>.
Then U - V = <u₁ - v₁, u₂ - v₂, ..., uₙ - vₙ>.
This method is consistent with adding the negative vector:
-V = <-v₁, -v₂, ..., -vₙ>
U + (-V) = <u₁ + (-v₁), u₂ + (-v₂), ..., uₙ + (-vₙ)> = <u₁ - v₁, u₂ - v₂, ..., uₙ - vₙ>.
Example (Algebraic):
Let U = <5, 3> and V = <2, 1>.
U - V = <5 - 2, 3 - 1> = <3, 2>
Let's check with adding the negative:
-V = <-2, -1>
U + (-V) = <5 + (-2), 3 + (-1)> = <5 - 2, 3 - 1> = <3, 2>.
The results match.
| Operation | Components | Result |
|---|---|---|
| U | <5, 3> | |
| V | <2, 1> | |
| U - V | <5 - 2, 3 - 1> | <3, 2> |
| -V | <-2, -1> | |
| U + (-V) | <5 + (-2), 3 + (-1)> | <3, 2> |
Subtracting vectors is a fundamental operation in linear algebra used in various applications, from physics (calculating relative velocity) to computer graphics (determining the vector between two points). The key is to remember that subtraction is the addition of the opposite vector.
