10.5#17

Hi Dr. Taylor. I am having trouble with getting the correct answer on this problem. I took the dot product of the two normal vectors and got -6. I took the magnitude of the two vectors and got the square root of 21 and the square root of 26. I then took the inverse cosine of -6 divided by the magnitudes. I checked my answer with multiple people and said that my calculations were correct. I am not sure what I am doing wrong. Let me know if you can help! Thank you!

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yes, that's a confusing and frustrating situation, and you're doing the right calculation--almost!
Let's imaging you're looking at the two planes edge on. You should see two lines, something like this:

and note that there are two angles between these two lines, one acute labeled as 𝛂 and the other one obtuse, and labeled as 𝛃.  They satisfy 𝛂+𝛃=π radians or 180 degrees. By convention, the angle between the two planes is take to be the acute angle 𝛂.  Similarly, let's look at the normal vectors for these planes. I've drawn the two normal vectors you've used in black, but for every normal vector n the vector -n is also a normal vector, which I've drawn in red.  The angles 𝛂 and 𝛃 show up between these normal vectors again, and you've computed 𝛃. What you need though is 𝛂, which you can compute as cos^(-1)(6/(√21√26)).