Distance between a point and a plane?

Question

How do I find the distance between a point, say for example (1,1,1) to a plane, say x+z=0? (Please show work)

Answer 1

Find the distance between a point, say for example P(1,1,1) to a plane, say x + z = 0.

Use the distance formula. The distance between a point
(x0, y0, z0) and the plane ax + by + cz + d = 0 is given by

distance = | a*x0 + b*y0 + c*z0 + d | / √(a² + b² + c²)
_____________

For the problem at hand we have:

distance = | 1*1 + 0*1 + 1*1 + 0 | / √(1² + 0² + 1²)
distance = | 1 + 0 + 1 + 0 | / √(1 + 0 + 1)
distance = 2/√2 = √2

Answer 2

JP L - This looks like an integral, but it cannot be! What will you integrate, and why? Here the quantity in question is distance, which dpends on discrete points, not a continuos region! Jerome - the point of rotation changes. Also, the amount of rotation is an AP. This is definitely not a ray sweeping a circle. How did you get to this conclusion? If Jerome is right, please explain! If he is not, maybe using complex points as a rotating arrow in a plane would be helpful. The LHS of the equaion will have terms like a1 - a2, a2-a3 and such so that on addition, every intermediate term will be cancelled off and we would get a1 - a182. If teh RHS can then be simplified and evaluated, you would get your answer.

Answer 3

the distance between a point (x1,y1,z1)and a plane ax+by+cz+d=0
is

|ax1+by1+cz1+d| / sqroot(a^2+b^2+c^2)

so the answer is

|1+1| / sqrt(1+1)
=1/1
=1