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The Angle Between Planes - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Which of these planes are parallel and which are perpendicular? Let’s take a quick look. I’ve named the planes M1, M2 and M3. Remember that planes are parallel if the normal vector of 1, is a scalar multiple of the normal vector of another. Let’s write down what the normal vectors are.

I’ll call n1 the normal for the first plane. It’s <2, -1, -3>. N2, the normal vector for this guy, <-4, 2, 6>. N3, the normal for the third plane <2, -8, 4>. Are any of these a scalar multiple of the other? Well looking at this list, I’m noticing that 2 times 2 is 4, 1 times 2 is 2, 3 times 2 is 6. If I multiply by -2, I can get this vector to become this vector. -2 times n1 is going to be <-4, 2, 6>. That’s exactly this guy. So n2 is -2 times n1, that means that plane m1 is parallel to plane m2.

Now what’s the relationship between these two planes which are parallel and this third one? Well it doesn’t actually appear that, well this guy is definitely not a scalar multiple of this guy. Because we would have to multiply the 2 times 1 to get 2, but we can’t multiply this by 1 to get -8. They’re not parallel for sure. But they could be perpendicular, and the way to test for that is to take the dot product.

Let’s take the dot product of n1 and n3. That’s going to be 2 times 2 which is 4, plus -1 times -8, 8. Plus -3 times 4, that’s -12. That is 0. That means that plane 1, M1 is perpendicular to M3. Since M1 is parallel to M2, M2 is also perpendicular to M3. That’s how the planes are related.

Remember the test for parallel planes is, one normal vector will be a scalar multiple of the other and the test for perpendicular is the dot product of the normal vectors will be zero.

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