A question on finding the intersecting line between two planes

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.

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1 Answer

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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Thank you for your answer. I realize why we assume $z=0$ equation-wise, but I didn't really get the vector-space explanation. If by plane they mean the $z$-axis, the intersecting line does NOT always have to intersect it (in case of one of the planes being parallel to the $y$-axis for instance). Then you have to try with either $x=0$ or $y=0$ like you said. If there really isn't anything more to it, than I guess the problem here is the vague(incorrect) explanation in my math book - am I correct? – Milosz Wielondek Mar 16 '11 at 11:24

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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. – Grumpy Parsnip Mar 16 '11 at 11:28
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