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The distance formula for points in space

The formula for the distance between two points in the plane extends to points in space, but why? This interactive figure will illustrate why the distance formula is the way it is in space. You can drag points A and B. Notice that when you hover over a point, arrows appear showing you how the point will move. Click on the point without dragging to change how the point moves.
Developed for use with Thomas' Calculus, published by Pearson.