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# What is the formula for Euclidean distance ?

Euclidean distance measures the distance between two vectors of length p denoting p traits of various observations and is a specific example of Mahalanobis distance with an identity covariance matrix (ie uncorrelated traits).

ED = for vectors, observations with vectors $$x_text{i} = (x_text{1i}, ..., x_text_{pi})text{T}$$ and $$x_text{j} = (x_text{1j}, ..., x_text_{pj})text{T}$$ equals $$\sqrt{(x_text{i} - x_text{j})^text{T}(x_text{i} - x_text{j})}$$

This can be written in long hand as $$\sqrt{(x_text{1i}-x_text{1j})text{2} + .. + (x_text{pi}-x_text{pj})text{2}}$$

The Euclidean distance is the distance on a graph between two points. This is easily seen in two dimensions since by Pythagoras's theorem the distance (hypotenuse) between two points (x11, x21) and (x12, x22) equals the square root of the squared difference in x and y co-ordinates = square root of (x11-x12)(x11-x12) + (x21-x22)(x22-x21). See [attachment:euclide.bmp here.]