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.]