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Distance is independent of direction because of the symmetry property of the metric. The formula $d(x,y) = d(y,x)$ shows that swapping the order of the points does not change the distance value.

Conditions: $x$ and $y$ are points in the metric space.

Distance is independent of direction because of the symmetry property of the metric. The formula $d(x,y) = d(y,x)$ shows that swapping the order of the points does not change the distance value.

Conditions: $x$ and $y$ are points in the metric space.

The video uses absolute value because simple subtraction can yield a negative result depending on the order of the operands. Distance must be non-negative ($d(x,y) \ge 0$).

Conditions: The points are real numbers on a number line.; The distance function must satisfy the non-negativity property.

The video uses absolute value because simple subtraction can yield a negative result depending on the order of the operands. Distance must be non-negative ($d(x,y) \ge 0$).

Conditions: The points are real numbers on a number line.; The distance function must satisfy the non-negativity property.