Finding the distance between a point and a plane using dot products

[haackeDc]

I know how I would be able to do this using projection, but am not so sure with dot products.

Do I dot the normal vector with an imaginary point and then figure something out from there?

If the normal is a= <a1,a2,a3>
and the random point is (p1,p2,p3)

If I dot them, I would get a1p1 + a2p2 + a3p3 = 0

And then solve for this point?

I don't see how to do it from here, could anyone help me?

 

[conquest]

(0, 0, 0) solves this!

I think though that you want to find the distance between any given point and any given plane. I think it should work if you shift the plane into the origin ( by setting the euqaiton for the plane to zero). Then shift your point accordingly. then you dot the vector going to the point into the unit normal vector to the plane.

 

[HallsofIvy]

If the plane is Ax+ By+ Cz= D and the point is (x_0, y_0, z_0), you know that <A, B, C> is a normal vector to the plane and so x= At+ x_0, y= Bt+ y_0, z= Ct+ z_0 is a line that passes through (x_0, y_0, z_0) and is perpendicular to the plane. Replace x, y, and z in the equation of the plane with those to solve for the point where the normal line passes through the plane and find the distance from (x_0, y_0, z_0) to that point.

 

[haackeDc]

If the plane is Ax+ By+ Cz= D and the point is (x_0, y_0, z_0), you know that <A, B, C> is a normal vector to the plane and so x= At+ x_0, y= Bt+ y_0, z= Ct+ z_0 is a line that passes through (x_0, y_0, z_0) and is perpendicular to the plane. Replace x, y, and z in the equation of the plane with those to solve for the point where the normal line passes through the plane and find the distance from (x_0, y_0, z_0) to that point.

Question resolved. Thanks for your help everyone.