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Unformatted text preview: Math 251, Spring 2008, Quiz 1 Solutions
Describe the region defined by x = y = z. How about x = y = z 2 ? What if you had just x = z 2 . Draw a picture in all three cases to get full credit. This part is worth 6 marks. We did in recitation last time that b - proja b is orthogonal to a because projection of a vector b onto a gives us the component of b in the direction of a. We'll try an analogue of that for cross products. ab ab Let a, b be two given vectors. Consider ppa,b c = c - (c. |ab| ) |ab| . What kind of a vector is this? Draw a picture and describe your answer in words. You neednt prove anything here. This part is worth 4 marks. x = y = z is a straight line. -10 -5 -10 -5 0 0 5 5 0 5 10 -5 10 -10 Figure 1: x = y = z x = y = z 2 is a 3D parabola. Its projection onto the xy plane is a straight line. 1 100 75 50 -10 25 -5 -5 0 0 0 5 5 10 10 -10 Figure 2: x = y = z 2 x = z 2 is a parabolic cylinder.
10 5 10 0 00 -5 -5 -10 5 25 50 -10 75 100 Figure 3: x = z 2 For the second part of the problem, the thing to realize is that a b gives a vector perpendicular to both a and b. Taking dot product of c
2 with that gives the projection of c in that direction. When you subtract this projection from c, you would get a vector that lies in the plane containing a, b. So this idea is the extention of the projection of one vector onto another. 3 ...
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