Date: Tue, 24 Jan 95 13:55:29 CST
From: rusin (Dave Rusin)
To: richards@sequent.com
Subject: Re: Points and Planes (simple question)

In article <1995Jan24.172610.12417@sequent.com> you write:
>
>You are given five points P1, P2, P3, P4 and P5 in three dimensional space.
>P1, P2 and P3 define a plane in the three dimensional space.
>
>If you were to draw a line through P4 and perpendicular to the plane where 
>would this line intersect the plane?
To get the direction perpendicular to the plane use the cross product: let
v = (P3-P1) cross (P2 - P1). Divide this by its length so that  v  is a
unit vector.
Now look at the vector  P4-P1 joining    P1 to P4. It can be decomposed
into a vector perpendicular to the plane plus one parallel to it. You
already know the direction of that first part (=v); its length is the
dot product  c=v dot (P4-P1). So c(P4-P1) is the normal component, and
P4- c(P4-P1) is the point you want.
 
>If you were to draw a line from P4 to P5 where would this line intersect the
>plane (if at all)?
This time decompose the vector (P5-P4). Again you compute the length  c'  of
the part perpendicular to the plane. Then (assuming  c'  isn't zero, i.e.,
P5-P4 is not parallel to the plane) the vector  (P5-P4).(c/c')  is a vector
pointing in the same direction as  P5-P4, but whose length in the 
direction perpendicular to the plane is the same as the distance from  P4
to the plane. So P4 + (P5-P4)(c/c') lies in the plane. 

This is standard Calc-3 stuff; look at the middle or end of the calculus
book used in your local university.

dave

==============================================================================


From: "Richard Shields (richards)" <richards@sequent.com>
To: rusin <rusin@math.niu.edu>
Subject: Re: Points and Planes (simple question)
Date: Wed, 25 Jan 95 09:12:00 PST

Thanks for your assistance.  It has been almost 10 years since I took Calc 
III and I haven't used any of it since (how quickly we forget what we don't 
use).  The directions you gave worked very nicely except for a one small 
error in the formula.  I believe the point in the plane in the first 
question should be P4 - cv.  Again, thanks for your help.

Thanks,
Rich

[my message, above, was attached.]