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The Angle Between Planes - Problem 2

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.

How do you find the angle between two planes? Here we have two planes; M1 and M2. M1 is given by the equation 2x plus 2y minus z equals 10, and M2 is 6x minus 3y plus 2z equals 24. First, to find the angle between planes you want to find the angle between their normal vectors.

Let’s observe that for plane 1, the normal vector is going to be <2, 2, -1>. And for plane 2, the normal vector is going to be <6, -3, 2>. How do you find the angle between two vectors? You take the dot product. Remember the formula cosine theta equals n1.n2 over the product of their magnitudes.

First thing I want to do is calculate their magnitudes. The magnitude of n1, 2² or 4 plus 4 plus 1. Square root of that. That’s the square root of 9, which is 3. And the magnitude of n2 is 6² or 36 plus 9 plus 4. The square root of 36 plus 9 plus 4. That’s 13 plus 36, 49 and root 49 is 7. This is going to be n1.n2. 2 times 6, 12, plus -6, plus -2, over 3 times 7. This is 12 times 8 which is 4 so I have 4 over 21.

Cosine of the angle between these two vectors is 4 over 21, so theta is going to be inverse cosine of 4 over 21. Let’s do that on a calculator. Inverse cosine of 4 over 21, I get 79 degrees approximately. So the angle between the two planes is 79 degrees.

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