Math 241, Quiz 1. 8/31/12.
Name:
• Read problems carefully. Show all work. No notes, calculator, or text.
• There are 15 points total.
1. §12.1, #9 b (5 points): Determine whether the points
D = (0, −5, 5), E = (1, −2, 4), F = (3, 4, 2)
lie on a straight line.
Solution: It suffices to calculate the distances between these points:
√
√
√
√
√
√
d(D, E) = 1 + 9 + 1 = 11, d(E, F ) = 4 + 36 + 4 = 2 11, d(F, D) = 9 + 81 + 9 = 3 11.
Now, since d(D, E) + d(E, F ) = d(F, D), we see that the points lie on a line.
−−→
−→
−−→
Alternative: Consider the vectors DE = h1, 3, −1i and EF = h2, 6, −2i. Since 2DE =
−→
−−→
−→
EF , we have DE k EF . These vectors share the point E, so the points E, F , D are
collinear.
2. §12.1, #11 (5 points): Find an equation of the sphere with center (1, −4, 3) and radius 5.
What is the intersection of this sphere with the xz-plane?
Solution: The equation of the sphere is
(x − 1)2 + (y + 4)2 + (z − 3)2 = 25.
To obtain its intersection with the xz-plane, set y = 0:
(x − 1)2 + (z − 3)2 = 9, y = 0.
This is a circle in the xz-plane with radius 3 and center (1, 0, 3).
3. §12.1, #33 (5 points): Write an inequality to describe the region between the yz-plane and
the vertical plane x = 5.
Solution: 0 < x < 5.