By definition, the absolute value of the difference of two vectors is the magnitude of the difference. This is a scalar quantity representing the length or size of the resulting difference vector.
Understanding Vector Subtraction and Magnitude
When we talk about subtracting two vectors, say vector A and vector B, the result is a new vector, C = A - B. This resulting vector C has both a magnitude (length) and a direction.
Unlike scalar subtraction where the absolute value of a difference |a - b| is simply the non-negative difference between two numbers, for vectors, the concept of "absolute value" directly corresponds to the magnitude of the resulting difference vector.
As highlighted by the reference: "By definition, the absolute value of the difference of two vectors is the magnitude of the difference. This is a scalar whose value is the same for vector A-B as for vector B-A." This is a crucial point:
- A - B results in a vector.
- B - A results in a vector pointing in the exact opposite direction of A - B.
- However, the magnitude of (A - B) is equal to the magnitude of (B - A).
This magnitude is always a non-negative scalar value.
Why Magnitude?
In mathematics and physics, the "size" or "length" of a vector is its magnitude. When extending the concept of "absolute value" (which represents distance from zero or size) to vectors, magnitude is the natural equivalent. It gives us a single number representing the scale of the vector difference, irrespective of its direction.
Calculating the Magnitude of the Difference
To find the magnitude of the difference between two vectors A and B:
- First, calculate the difference vector C = A - B. If A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) in three dimensions, then C = (a₁ - b₁, a₂ - b₂, a₃ - b₃).
- Then, calculate the magnitude of vector C. The magnitude of a vector (c₁, c₂, c₃) is given by the formula:
||C|| = √((c₁)² + (c₂)² + (c₃)²)
This formula is derived from the Pythagorean theorem and applies regardless of the number of dimensions.
Example: Vector Subtraction and Magnitude
Let's consider two 2D vectors:
- A = (4, 5)
- B = (1, 2)
-
Calculate the difference vector A - B:
A - B = (4 - 1, 5 - 2) = (3, 3) -
Calculate the magnitude of the difference vector (3, 3):
Magnitude = ||(3, 3)|| = √(3² + 3²) = √(9 + 9) = √18
So, the absolute value of the vector subtraction A - B is √18.
Let's check for B - A:
-
Calculate the difference vector B - A:
B - A = (1 - 4, 2 - 5) = (-3, -3) -
Calculate the magnitude of the difference vector (-3, -3):
Magnitude = ||(-3, -3)|| = √((-3)² + (-3)²) = √(9 + 9) = √18
As the reference states, the magnitude is indeed the same for A - B and B - A, even though the vectors themselves point in opposite directions.
| Vector Operation | Resulting Quantity | Type |
|---|---|---|
| Vector Subtraction | A vector | Vector |
| Magnitude of Vector | Its length/size | Scalar |
| Abs Value (Vector Subtraction) | Magnitude of the difference vector | Scalar |
In summary, the "absolute value" in the context of vector subtraction specifically refers to the length or magnitude of the resulting difference vector, which is a non-directional scalar quantity.
