Let A(3,2,-1), B(5,0,-2), C(3,3,0) and D(4,-6,8) be four points in the plane,Donate the plane containing A, B, and C.Use vector methods to solve the following.
(a)Find a unit vector perpendicular to the plane.
(b)WITHOUT finding the equation of the plane , calculate the shortest distance between D and the plane
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Expert's answer
2020-03-03T16:37:31-0500
Let α be the plane containing points A, B, and C. The vector n is a unit vector perpendicular to the plane. AB=(2,−2,−1),AC=(0,1,1),AD=(1,−8,9).
a) The unit vector n can be found as a cross-product of vectors AC and AB normalized by its length.
n=∣AB×AC∣AB×AC .
AB×AC=∣∣i20j−21k−11∣∣=(−1,2,2).
∣AC×AB∣=(−1)2+(2)2+(2)2=3.
Thus, n=31(−1,2,2).
b) The shortest distance DD′ between D and the plane can be found as a projection of vector AD onto the n direction. Namely,
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