Distance from a point to a plane

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MCV4U1-UNIT NINE-LESSON FIVE
Lesson Five: Distance Formulas
Distance from a point to a line in  2 :
P

d  proj n OP

n   A, B
Ax+By+c=0
O

 OP  n
Ax1  By 1  C
d  proj n OP 
 
n
A2  B 2
Distance from a point to a line in 3 :
Q
d

d
θ
P
sin  
d

PQ

 d  PQ sin 

 

and recall magnitude of cross product ...... d  PQ  d PQ sin 


d  PQ
 sin    
d PQ


  d  PQ 
d  PQ    
 d PQ 




d  QP
d 

d
We can drop the absolute value bars on sinθ, b/c
here, θ will always be between 0° and 180°.
MCV4U1-UNIT NINE-LESSON FIVE
Distance from a point to a plane:
The distance from a point to a plane is.......
 
AX  n
d

n
where A is the external point and X is a point on the plane.
Example: Find the distance from the point (-1,0,3) to the plane x-3y+2z-1=0
 
AX  n
d

n
A=(-1,0,3), X=any point on the plane.... let y=0,z=0, then
x=1, so X=(1,0,0)
d 
2,0,3  1,3,2
14


AX  2,0,3 ,
4
14


n  1,3,2
4
4 14 2 14


14
7
14

n  14
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