Distance

Distance is one of the two fundamental components of vectors.

Contents

  1. Distance
    1. Pythagorean distance
    2. Euclidean distance

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 dot product of a vector by itself. For vector a⃗ with i components, this could be written out as Σ(ai2). 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.

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