= Solution
For rows $x_i,x_j$, three useful choices are <Euclidean distance>, <standardized Euclidean distance>, and <Mahalanobis distance>:
$$
d_E(i,j)=\sqrt{\sum_{r=1}^p(x_{ir}-x_{jr})^2},\qquad
d_S(i,j)=\sqrt{\sum_{r=1}^p\frac{(x_{ir}-x_{jr})^2}{s_r^2}},\qquad
d_M(i,j)=\sqrt{(x_i-x_j)^TS^{-1}(x_i-x_j)}.
$$
The Euclidean choice is simple, rotation invariant and appropriate when the original coordinate scales carry comparable meaning. It is sensitive to scale changes, may be dominated by a high-variance variable and gives correlated or duplicate variables repeated influence. Squared Euclidean dissimilarity is sometimes used algorithmically, but removing the square root generally loses the <triangle inequality>.
The standardized Euclidean choice removes marginal units using fixed positive sample scales $s_r$. It gives equal marginal weight to variables, but ignores correlations and can magnify noise when an estimated <variance> is small. Constant variables require omission or an explicitly chosen positive external scale.
The Mahalanobis choice incorporates the full <positive-definite> sample covariance $S$, so it is <Euclidean distance> after whitening. It accounts for both scale and redundancy and is invariant under an invertible change of coordinates when $S$ is transformed consistently. However estimating $S$ requires enough observations and can be sensitive to extreme values; inversion is unstable near singularity and unavailable when $S$ is singular. With fixed positive scales or <positive-definite> $S$, all three square-root expressions are genuine <metrics> on the measurement space. Their different geometries encode different notions of which individuals should be considered similar.
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