Cross Product Proof: Proving Distance Formula for a Point Not on a Plane

[ProPatto16]

Homework Statement

Let P be a point not on the plane that passes through the points Q, R and S. show that the distance d from P to the plane is d = (|a.(bxc)|)/(|axb|)
where a = QR and b = QS and c = QP (those are lines between given two points)

Homework Equations

|axb| = (|a||b|sin\theta)n

|a||b| = (a.b)/cos\theta

distance eq for a line =

The Attempt at a Solution

no idea.
i struggle with proofs.
help please!

 

[lanedance]

how can a point pass through points? doesn't make sense...

 

[vela]

It's pretty straightforward to see what's going on geometrically. Start by thinking about what the vector \mathbf{a}\times\mathbf{b} is in relation to the plane.

 

[ProPatto16]

assuming P is arbitrary, then the vector a x b is the unit vector n parallel to plane of QRS.
say that distance d = vector PQ then a x b = vector n normal to plane. so then distance of P to the plane is the absolute value of the scalar projection of c onto n.

yeah?

 

[vela]

a x b isn't a unit vector; also you say it's both parallel and normal to the plane. But I think you have the basic idea.

 

[HallsofIvy]

how can a point pass through points? doesn't make sense...

"Let P be a point not on the plane that passes through the points Q, R and S."

A bit ambuguous but it is the plane that passses thorugh P, R, and S.

 

[ProPatto16]

oh i don't mean parallel to plane. my bad. a x b gives vector n normal to plane QRS. then PQ = c and distance d is absolute value of scalar projection of c onto n.