Register to reply

Distance formula for between a point and a plane

by ProPatto16
Tags: distance, formula, plane, point
Share this thread:
ProPatto16
#1
Mar3-11, 05:48 AM
P: 315
1. The problem statement, all variables and given/known data

Let P be a point not on a plane that passes through points Q, R, S. show that the distance, d, from P to the plane is:

d = |a.(b x c)| over |a x b| where a=QR, b=QS and c=QP

2. Relevant equations

definition of dot product is a.b=|a||b|cos(theta)
definition of cross product is a x b= |a||b|sin(theta)
triple scalar product is |a.(b x c)|=|b x c||a|cos(theta)

3. The attempt at a solution

putting point P above point S in plane, gives d = |PS| =|PQ|sin(theta) = |c|sin(theta)
theta is angle between |QP| and |QS| which is c and b so by definition of the cross product sin(theta)=|a x b| over |a||b| gives d = csin(theta) = |c||b x c| over |c||b| = |b x c| over |b|

to incorporate dot product i need a cos theta and the only relavant one i can figure is angle between QR and QS which is a and b. which wouldnt be in the plane. im not sure where to go from here??

thanks
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
ProPatto16
#2
Mar3-11, 05:19 PM
P: 315
i can do it with 2 points on a line if thats any help i will show it.

let P be a point not on the line that passes through Q and R. show that the distance d from the point P to the line L is

d = |a x b| over |a|

have line QR with point P drawn above and an arbitrary point S on the line where the perpendicular of P meets. let QR be vector a and QP be vector b.

the distance d is PS. which is also QPsin(theta) = |b|sin(theta). theta is angle between QP=b and QR=a therefore by definition of cross product, sin(theta) = |a x b| over |a||b|
and so d = |b|sin(theta) = |b||a x b| over |a||b| = |a x b| over |a|


but i dont know how to incorporate that into a plane with a dot product in the solution?
ProPatto16
#3
Mar5-11, 04:07 PM
P: 315
help?


Register to reply

Related Discussions
Please help to Find coordinates of a point formula Angle,Distance & first point known Differential Geometry 3
Create plane from 3 points and distance from plane to point General Math 5
Distance between point-plane & plane-plane Linear & Abstract Algebra 6
Distance from a point on a sphere to a point on a plane... Calculus 6