determine the equation of a plane, P3, that intersects the planes P1: x + y + 3z − 2 = 0 and P2: x − y + 2z = 0 in

Question

a point;
a line.

I know the normals are (1,1,3) and (1,-1,2) clearly the lines don't intersect because the normals are't multiples. So I don't know how to start the question

Answer 1

well, any old plane that is not parallel to the other two will intersect both at a point along the lines of intersection of those planes.
The line of intersection will be parallel to the cross product of or two normals
i j k
1 1 3
1-1 2
Find a plane that has that line of intersection as a normal and you have one that hits both planes at a point on their intersection.

Now to intersect both in the same line, your plane better include the line of intersection of the first two. Any old plane that includes that line will do (except either of the original two of course :)