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#deck:Math::Distance in n dimensions
#tags:math precalculus distance-n-dimensions
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How long is the diagonal of a box 3 by 4 by 12?	13: the floor diagonal is 5 (3-4-5), then sqrt(25 + 144) = 13 (5-12-13). Pythagoras twice.	formula
State the distance formula in three dimensions.	d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2): the plane formula with one more square.	formula
Why is the three-dimensional distance formula Pythagoras twice?	Once in the floor, for the floor diagonal; once standing up, with the floor diagonal and the vertical edge as legs. Substituting the first into the second collapses them into one sum of three squares.	why
Where does the three-dimensional formula use that the z-axis is perpendicular to the floor?	In the second right triangle: the angle between the floor diagonal and the vertical edge is right only because the vertical edge is perpendicular to every line in the floor.	why
How is the distance formula in n dimensions justified when nobody can draw a fourth axis?	By induction: the formula in n dimensions is the formula in n - 1 dimensions plus Pythagoras once more. The step is algebra on a sum of squares, which does not care how many terms it has.	why
What is the length of a vector v?	The distance from the origin to its tip: |v| = sqrt(v1^2 + ... + vn^2) = sqrt(v . v).	formula
Why is sqrt(v . v) a length and not just a number called one?	v . v is the sum of the squares of the coordinates, which the distance formula says is the squared distance from the origin to the point. The norm is defined in algebra; the geometry earns it the name.	why
Is 'the distance formula follows from |v| = sqrt(v . v)' a proof?	No. The dependence runs the other way: sqrt(v . v) is a length because of the distance formula. As a proof of the formula it assumes its conclusion in vector notation.	trap
What are the four properties a distance (a metric) must have?	Non-negative; zero exactly when the two points are the same; symmetric; and the triangle inequality d(P, R) <= d(P, Q) + d(Q, R).	formula
Which metric property of the Euclidean distance does not follow from the formula in one line?	The triangle inequality. Its proof goes through the Cauchy-Schwarz inequality; the other three are a sum of squares being non-negative, zero only when every term is, and blind to sign.	trap
Two points in four dimensions share their last coordinate. What does the formula reduce to?	The three-dimensional formula: the fourth square is 0 and drops out, as the plane formula reduces to |x2 - x1| for two level points.	formula
The distance between two points, in vector language?	The length of their difference: d(u, w) = |w - u|, because the differences of the coordinates are the coordinates of the difference.	formula
Sullivan writes two points as (x1, y1), (x2, y2); Wikipedia writes p = (p1, p2), q = (q1, q2). What does the subscript mean in each?	In Sullivan it numbers the point and the letter names the axis, so x2 is the x of the second point. In Wikipedia it numbers the axis and the letter names the point, so q1 is the first coordinate of q. Sullivan's x2 is Wikipedia's q1, not q2. The second convention scales to n axes.	trap