Find the minimum distance between the given point P and the given subspace W of ℝ³
P = (4, −1, 2), W = span({[−2, 3, −3]}) in ℝ³
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Answer:
Find the minimum distance between the given point P and the given subspace W of R³, P = (4, −1, 2), W = span({[−2, 3, −3]}) in R³
Step-by-step explanation:
Let [-2t, 3t, -3t] any vector from W. Then scuare of distance D2 = (4 + 2t)2 + (-1 - 3t)2 + (2 + 3t)2;
Derivative d(D2)/dt = 2(4 + 2t)·2 + 2(1 + 3t)·3 + 2(2 + 3t)·3 = 16 + 8t + 6 + 18t + 12 + 18t = 34 + 44t = 0;
t = - 17/22; Because d2(D2)/dt2 = 44 > 0 and we have minimum
D2 = (4 - 17/11)2 + (-1 + 51/22)2 + (2 - 51/22)2 = (27/11)2 + (29/22)2 + (-7/22)2 = 3806/222;
Dmin = √3806/22 ≈ 2.8
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