Science:Math Exam Resources/Courses/MATH253/December 2012/Question 02 (a)
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Question 02 (a) 

Let be the acute angle between the plane and the plane tangent to the surface at the point (1,2,5). Find . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The angle(s) between two planes is(are) given by the angle(s) between their respective normal vectors. Recall that there are two normal vectors for every plane. 
Hint 2 

Remember that the dot product between two vectors u and v satisfies where denotes the length of vector u and is the angle between u and v. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The angle between two planes is equal to the angle between their normal vectors. Using the hint, we remember that the angle between two vectors u, v satisfies where denotes the length of vector u. The normal vector to the plane is (4,2,1). To determine the vector normal to the surface at (1,2,5), we compute the gradient of the function and evaluate it at the point (1,2,5), Using the formula above, we compute , Notice that the sign of is negative, which implies an obtuse angle between the vectors. This may seem like a problem, but recall that every plane has two normal vectors which point in opposite directions. If we take the negative of either or the normal vector to the plane and use it in our calculation for , we get the positive result: 