Lecture 3. DISTANCES Math21a, O. KnillDISTANCE POINT-POINT (3D). If P and Q are two points, thend(P, Q) = |~P Q|is the distance between P and Q.DISTANCE POINT-PLANE (3D). If P is a point in space and Σ : ~n ·~x = d isa plane containing a point Q, thend(P, Σ) = |(~P Q) ·~n|/|~n|is the distance between P and the plane.DISTANCE POINT-LINE (3D). If P is a point in space and L is the line~r(t) = Q + t~u, thend(P, L) = |(~P Q) × ~u|/|~u|is the distance between P and the line L.DISTANCE LINE-LINE (3D). L is the line ~r(t) = Q + t~u and M is the line~s(t) = P + t~v, thend(L, M) = |(~P Q) · (~u ×~v)|/|~u ×~v|is the distance between the two lines L and M.DISTANCE PLANE-PLANE (3D). If ~n · ~x = d and ~n · ~x = e are two parallelplanes, then their distance is (e − d)/|~n|. Nonparallel planes have distance 0.EXAMPLESDISTANCE POINT-POINT (3D). P = (−5, 2, 4) and Q = (−2, 2, 0) are twopoints, thend(P, Q) = |~P Q| =p(−5 + 2)2+ (2 − 2)2+ (0 − 4)2= 5DISTANCE POINT-PLANE (3D). P = (7, 1, 4) is a point and Σ : 2x+4y+5z =9 is a plane which contains the point Q = (0, 1, 1). Thend(P, Σ) = |(7, 0, 3) · (2, 4, 5)|/|√45| = 29/√45is the distance between P and Σ.DISTANCE POINT-LINE (3D). P = (2, 3, 1) is a point in space and L is theline ~r(t) = (1, 1, 2) + t(5, 0, 1). Thend(P, L) = |(1, 2, −1) × (5, 0, 1)|/√26 = |(2, −6, −10)|/√26 =√140/√26is the distance between P and L.DISTANCE LINE-LINE (3D). L is the line ~r(t) = (2, 1, 4) + t(−1, 1, 0) andM is the line ~s(t) = (−1, 0, 2) + t(5, 1, 2). The cross product of (−1, 1, 0) and(5, 1, 2) is (2, 2, −6). The distance between these two lines isd(L, M) = |(3, 1, 2) · (2, 2, −6)|/√44 = 4/√44 .DISTANCE PLANE-PLANE (3D). 5x + 4y + 3z = 8 and 5x + 4y + 3z = 1 aretwo parallel planes. Their distance is
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