Rock D.

asked • 07/20/21

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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So, this takes some basic decoding of language. The subspace W is the span of a single vector in 3 dimensions, so it is simply a line in 3D space. In this case, since we're working within a Euclidean space, the minimum distance is simply the magnitude of the straight line segment perpendicular to W that passes through point P. Knowing this, it becomes much easier to conceptualize how to move forward. Remember two things: first, you can create a vector orthogonal to another by noting that the dot product of orthogonal vectors is zero; second, you can create find a line that passes through a point in a certain direction with the formula for the vector equation of a line. From here, it's simply a matter of calculating the length along this line from the point to W.
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07/20/21

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Inactive Tutor answered • 07/22/21

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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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