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)"
To: rusin
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.]