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| '''Euclidean distance''' in higher dimensional space is useful for normalization. | '''Distance''' is one of the two fundamental components of [[Calculus/VectorOperations|vectors]]. |
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| == 2 Dimensions == | == Pythagorean distance == |
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| In 2 dimensions, the distance between two Cartesian points is calculated by the '''Pythagorean theorem''': ''x^2^ + y^2^ = z^2^''. This is sometimes referred to as the '''Pythagorean distance'''. | In ''R^2^'' space, the distance between two Cartesian points is calculated by the '''Pythagorean theorem''': ''x^2^ + y^2^ = z^2^''. |
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| 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. |
The Pythagorean distance of a vector a⃗ is notated ''|a⃗|''. |
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| == n Dimensions == | == Euclidean distance == |
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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. | Euclidean distance is the expansion of Pythagorean distance into ''R^n^'' space. Although [[Calculus|vector calculus]] largely stays in ''R^3^'', this property is generalized for higher dimensions. |
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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 squared Euclidean distance is the sum of squares across all components of the vector. For vector a⃗ with ''i'' components, this could be written out as ''Σa,,i,,''. In [[LinearAlgebra|linear algebra]] however, the more conventional expression is ''x^T^x'' for a given vector ''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. | Taking the square root then gives the actual Euclidean distance, which for a vector a⃗ is notated ''||a⃗||''. |
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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 proven in ''R^3^'' trivially. For a vector a⃗ with X, Y, and Z components, use the Pythagorean theorem on the X and Y components to calculate a hypotenuse vector. Then use it again with that hypotenuse vector and the Z component. |
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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]]. | More generally, the Pythagorean theorem can be shown to hold in ''R^n^'' space. Suppose that ''x'' is ''[1 2 3]'' and ''y'' is ''[2 -1 0]''. [[Calculus/VectorOperations#Addition|Vector addition]] demonstrates that ''z = x + y = [1 2 3] + [2 -1 0] = [3 1 3]''. The Pythagorean theorem would then suggest that ''x^T^x + y^T^y = z^T^z''; and it is straightforward to demonstrate that both the left and right hand sides of the equation are ''19''. |
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| Distance of a vector is commonly notated as ''||z||''. | The Pythagorean theorem can be restated as ''x^T^x + y^T^y = (x+y)^T^(x+y)''. This formulation leads to the test for [[LinearAlgebra/Orthogonality#Vectors|orthogonality]]. |
Distance
Distance is one of the two fundamental components of vectors.
Pythagorean distance
In R2 space, the distance between two Cartesian points is calculated by the Pythagorean theorem: x2 + y2 = z2.
The Pythagorean distance of a vector a⃗ is notated |a⃗|.
Euclidean distance
Euclidean distance is the expansion of Pythagorean distance into Rn space. Although vector calculus largely stays in R3, this property is generalized for higher dimensions.
The squared Euclidean distance is the sum of squares across all components of the vector. For vector a⃗ with i components, this could be written out as Σai. In linear algebra however, the more conventional expression is xTx for a given vector x.
Taking the square root then gives the actual Euclidean distance, which for a vector a⃗ is notated ||a⃗||.
This can be proven in R3 trivially. For a vector a⃗ with X, Y, and Z components, use the Pythagorean theorem on the X and Y components to calculate a hypotenuse vector. Then use it again with that hypotenuse vector and the Z component.
More generally, the Pythagorean theorem can be shown to hold in Rn space. Suppose that x is [1 2 3] and y is [2 -1 0]. Vector addition demonstrates that z = x + y = [1 2 3] + [2 -1 0] = [3 1 3]. The Pythagorean theorem would then suggest that xTx + yTy = zTz; and it is straightforward to demonstrate that both the left and right hand sides of the equation are 19.
The Pythagorean theorem can be restated as xTx + yTy = (x+y)T(x+y). This formulation leads to the test for orthogonality.