SOLUTION: points A(3,-5,1) and B(2,1,2) and C (1,-3,3) lie on a plane. Find the exact distance from X (2,-1,3) to the plane

Algebra ->  Vectors -> SOLUTION: points A(3,-5,1) and B(2,1,2) and C (1,-3,3) lie on a plane. Find the exact distance from X (2,-1,3) to the plane       Log On


   

Question 882303: points A(3,-5,1) and B(2,1,2) and C (1,-3,3) lie on a plane. Find the exact distance from X (2,-1,3) to the plane
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
First find the equation of the plane.
AB=(-1,6,1)
BC=(-1,-4,1)
n=ABxBC=%28matrix%283%2C3%2Ci%2Cj%2Ck%2C-1%2C6%2C1%2C-1%2C-4%2C1%29%29

n=10i%2B0j%2B10k
n%2A%28matrix%281%2C3%2Cx-3%2Cy%2B5%2Cz-1%29%29=0
10%28x-3%29%2B0%28y%2B5%29%2B10%28z-1%29=0
10x-30%2B10z-10=0
10x%2B10z=40
highlight_green%28x%2Bz=4%29
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Now that you have the equation of the plane,
L=abs%28ax%5B0%5D%2Bby%5B0%5D%2Bcz%5B0%5D%2Bd%29%2Fsqrt%28a%5E2%2Bb%5E2%2Bc%5E2%29
where the plane is defined by ax%2Bby%2Bcz%2Bd=0 and the point is
(x%5B0%5D,y%5B0%5D,z%5B0%5D)
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.
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a=1
b=0
c=1
d=-4
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.
.
L=abs%281%282%29%2B1%283%29-4%29%2Fsqrt%281%5E2%2B0%5E2%2B1%5E2%29
L=abs%282%2B3-4%29%2Fsqrt%282%29
L=1%2Fsqrt%282%29
highlight%28L=sqrt%282%29%2F2%29