Find the parametric equation for the line...

Question

Find the parametric equation for the line in which the planes x + y + z = 0 and 2x + y + −z = 2 intersect. What is the distance from point P₀(0, 0, 1) to this line?

Answer 1

Here are some suggestions.

To find a point that lies on the line of intersection, find a point that lies on both planes by letting z=0 (for example) and solve the remaining system for x and y. That point has coordinates (2,-2,0), The direction vector for the line will be orthogonal to the normals for the planes. To find that direction vector find the cross product of the normals. Having a point (2,-2,0) and a direction vector allows you to write either the parametric or symmetric equations for the line of intersection.

To find the distance from P₀=(0,0,1) to the line, evaluate ||u x v|| / ||v|| where u is the vector from P₀ to (2,-2,0) and v is the direction vector of the line, i.e. the cross product of the normals. Hopefully the above gets you pointed in the right direction.