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According to my math book, in order to find the intersecting line between two planes we need to:

  1. Find the vector product of the direction normals of the two planes
  2. Write the equations of the planes in Cartesian form.
  3. Assume that $z=0$ since the line has to intersect this plane.
  4. Solve simultaneously for a point on the line and write down a vector equation of the line

What is meant by point 3? I'm having problems visualizing it. Please be explicit.

1 Answers 1

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To find the line of intersection, you need two pieces of information. The direction of the line as a vector and a point on the line. Step 3 and the first part of step 4 is about locating a single point on the line of intersection. Since you are only looking for a point, you might as well assume $z=0$, to reduce your two plane equations to two equations in two unknowns which can be solved simultaneously. Now this might not always work. Sometimes the line of intersection happens to be parallel to the $z=0$ plane. In that case you could try $y=0$ or $x=0$. (One of these is sure to work.)

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    I agree with you except that $z=0$ represents the $xy$-plane, not the $z$-axis. Still, if your book says that $z=0$ always works, then it is not quite right.2011-03-16