Vectors have the same magnitude when the sum of the squares of their respective components is equal, even if their components are different.
Understanding vector magnitude is fundamental in many areas of science and engineering. The magnitude of a vector, also known as its length or norm, represents the overall "size" or strength of the vector, irrespective of its direction.
Calculating Vector Magnitude
For a vector in two dimensions, say $\mathbf{v} = (v_x, v_y)$, the magnitude is calculated using the Pythagorean theorem:
Magnitude $(|\mathbf{v}|) = \sqrt{v_x^2 + v_y^2}$
In three dimensions, for a vector $\mathbf{w} = (w_x, w_y, w_z)$, the magnitude is:
Magnitude $(|\mathbf{w}|) = \sqrt{w_x^2 + w_y^2 + w_z^2}$
This concept extends to any number of dimensions. The key is that the magnitude is the square root of the sum of the squares of the vector's components.
How Different Vectors Can Share Magnitude
As the reference points out, even if two vectors have different components, they can have the same magnitude if the squared sum of their respective components is identical. This is because the order and signs of the components can change, but as long as the sum of their squares remains the same, the final square root (the magnitude) will be the same.
Consider these examples:
Example 1: Different Order of Components
The reference provides the example of vectors (3, 4) and (4, 3). Let's calculate their magnitudes:
-
Vector (3, 4):
- Sum of squared components = $3^2 + 4^2 = 9 + 16 = 25$
- Magnitude = $\sqrt{25} = 5$
-
Vector (4, 3):
- Sum of squared components = $4^2 + 3^2 = 16 + 9 = 25$
- Magnitude = $\sqrt{25} = 5$
As you can see, despite having different components, the sum of their squares is the same (25), resulting in the same magnitude (5).
Let's visualize this in a table:
| Vector | Component 1 | Component 2 | Squared Component 1 | Squared Component 2 | Sum of Squared Components | Magnitude |
|---|---|---|---|---|---|---|
| (3, 4) | 3 | 4 | $3^2 = 9$ | $4^2 = 16$ | $9 + 16 = 25$ | $\sqrt{25} = 5$ |
| (4, 3) | 4 | 3 | $4^2 = 16$ | $3^2 = 9$ | $16 + 9 = 25$ | $\sqrt{25} = 5$ |
Example 2: Different Signs of Components
Consider vectors (3, 4), (-3, 4), (3, -4), and (-3, -4).
- Vector (3, 4): Magnitude = $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
- Vector (-3, 4): Magnitude = $\sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
- Vector (3, -4): Magnitude = $\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
- Vector (-3, -4): Magnitude = $\sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
In these cases, the negative signs disappear when the components are squared, resulting in the same sum of squares and therefore the same magnitude.
In summary, vectors have the same magnitude not by having the same components, but by having the same value for the square root of the sum of the squares of their components.
