if y=[6;-3;7] and u=[-6,-2,-3] .compute the distance d , from y to the line thro

Question

if y=[6;-3;7] and u=[-6,-2,-3] .compute the distance d , from y to the line through u and the origin. plz show work...d=?

Explanation / Answer

Let L = line that passes through origin (0,0,0) and point U(7,2,2) L: (x,y,z) = (7t, 2t, 2t) Direction vector of line L is < 7, 2, 2 > Let point P(7t, 2t, 2t) be the point on line L that is closest to Y(-1,5,10) Vector from point Y to point P = < 7t, 2t, 2t > - < -1, 5, 10 > = < 7t+1, 2t-5, 2t-10 > Since P is closest point to Y on line L, then vector YP is perpendicular to line L Therefore, dot product of direction vectors = 0 < 7t+1, 2t-5, 2t-10 > • < 7, 2, 2 > = 0 49t + 7 + 4t - 10 + 4t - 20 = 0 57t - 23 = 0 57t = 23 t = 23/57 Length of vector YP = distance from Y to P = shortest distance from Y to line L = | < 7t+1, 2t-5, 2t-10 > | = | < 7(23/57)+1, 2(23/57)-5, 2(23/57)-10 > | = 1/57 | < 7*23+57, 2*23-5*57, 2*23-10*57 > | = 1/57 | < 218, -239, -524 > | = 1/57 v(218² + 239² + 524²) = 10.80367

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