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⇤ ← Revision 1 as of 2024-01-27 22:49:37
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← Revision 2 as of 2024-01-29 19:53:04 ⇥
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Comment: Euclidean distance fix
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| For a singular vector ''x'', the distance is the sum of each components' absolute value. If ''x'' is ''[1 2 3]'' or ''[-1 -2 -3]'', the length of ''x'' is 6. | For a singular vector ''x'', the distance is the sum of each components' absolute value. If ''x'' is ''[1 2 3]'' or ''[-1 -2 -3]'', the distance of ''x'' is 6. Note that distance of a vector ''x'' is notated as ''||x||''. |
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| For multiple vectors, the total distance is the sum of each vectors' distance. Continuing with the example for ''x'', if ''y'' were ''[2 -1 0]'', then their total distance would be 14 + 5 = 19. | The Pythagorean theorem continues to hold in higher dimensions. Note that the theorem calls for squared distances: ''x^2^''. In other words, the squared distance of ''x'' is the sum of each component squared. For either of those ''x'' vectors, the squared distance is 14. Recall though that in matrix notation, multiplying two vectors creates a matrix; true vector multiplication is notated as ''x^T^x''. |
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| The total distance can also be thought of as the distance of the singular vector ''z'' resulting from ''x+y''. If ''x'' were ''[1 2 3]'' and ''y'' were ''[2 -1 0]'', then ''z'' would be ''[3 1 3]'' and the distance of ''z'' is 19. | Plugging this into the Pythagorean theorem then, if ''y'' were ''[2 -1 0]'' (with squared distance of 5), then the squared distance of ''z'' is 19. The actual distance is √19. |
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| The Pythagorean theorem continues to hold in higher dimensions. Importantly though, the ''squared'' distance of a singular vector ''x'' is the sum of each components' square. For either of the above ''x''s, the squared distance is 14. This can be notated as ''x^T^x''. For two vectors ''x'' and ''y'', the Pythagorean theorem is written as ''x^T^x + y^T^y = (x+y)^T^(x+y)''. | This can be double checked using the given values of ''x'' and ''y'', because vectors can be added directly. ''x + y = z'', so ''[1 2 3] + [2 -1 0] = [3 1 3]''. And with that calculated ''z'', it's clear again that the squared distance of ''z'' is 19. |
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| Because it is simple to sum each vectors' distance, the Pythagorean theorem isn't used to compute total distance in higher dimensions. Rather, it leads to the proof for [[LinearAlgebra/Orthogonality#Vectors|orthogonality]]. Distance of a vector is commonly notated as ''||z||''. |
For two vectors ''x'' and ''y'', the Pythagorean theorem can be written as ''x^T^x + y^T^y = (x+y)^T^(x+y)''. This formulation leads to the test for [[LinearAlgebra/Orthogonality#Vectors|orthogonality]]. |
Distance
Euclidean distance in higher dimensional space is useful for normalization.
Contents
2 Dimensions
In 2 dimensions, the distance between two Cartesian points is calculated by the Pythagorean theorem: x2 + y2 = z2. This is sometimes referred to as the Pythagorean distance.
When this concept is expanded to complex numbers (but still in 2 dimensions), a common notation is |x - y| to emphasize that the distance must be normalized to an absolute value.
n Dimensions
For a singular vector x, the distance is the sum of each components' absolute value. If x is [1 2 3] or [-1 -2 -3], the distance of x is 6. Note that distance of a vector x is notated as ||x||.
The Pythagorean theorem continues to hold in higher dimensions. Note that the theorem calls for squared distances: x2. In other words, the squared distance of x is the sum of each component squared. For either of those x vectors, the squared distance is 14. Recall though that in matrix notation, multiplying two vectors creates a matrix; true vector multiplication is notated as xTx.
Plugging this into the Pythagorean theorem then, if y were [2 -1 0] (with squared distance of 5), then the squared distance of z is 19. The actual distance is √19.
This can be double checked using the given values of x and y, because vectors can be added directly. x + y = z, so [1 2 3] + [2 -1 0] = [3 1 3]. And with that calculated z, it's clear again that the squared distance of z is 19.
For two vectors x and y, the Pythagorean theorem can be written as xTx + yTy = (x+y)T(x+y). This formulation leads to the test for orthogonality.