Show that the points (2,0,1), (0,4,-3), (-2,5,0) are non-collinear. Hence the equation of plane passing through them.
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Expert's answer
2020-11-17T16:25:26-0500
LetA=(2,0,1),B=(0,4,3),C=(−2,5,0)IfA.(BC)=0,the three points are collinear.∣∣20−2045130∣∣=2∣∣4530∣∣+∣∣0−245∣∣=2(−15)+8=−30+8=−22SinceA.(B×C)=0,the three points arenot collinear.Also, since we are not given a normalvector, we need to find one by takingthe cross product of the displacement vectorufromAtoBand thedisplacement vectorvfromBtoC.Then we obtain a vectornwhich isnormal (orthogonal) to each of the originalvectors (and thus orthogonal to the plane).AB=(0,4,3)−(2,0,1)=(−2,4,2)BC=(−2,5,0)−(0,4,3)=(−2,1,−3)u×v=(−2,4,2)×(−2,1,−3)u×v=∣∣i−2−2j41k2−3∣∣=i(−12−2)−j(6+4)+k(−2+8)=−14i−10j+6kn=(−14,−10,6),a=(2,0,1)Hence the equation of the plane is(r−a)n=0−14(x−2)−10(y−0)+6(z−1)=0−14x−10y+6z=−22
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