1. What did you find most difficult about 12.6?
I believe the hardest part of section 12.6 is knowing the shapes of each of the different quadric surfaces. The names and shapes of all the different quadric surfaces are similar and hard to distinguish.
2. What did you find most interesting about 12.6?
What I found interesting was the new cylinders shown in this section. We are all so used to working solely with the right-circular cylinders, however in this section we are introduced to elliptic cylinders, hyperbolic cylinders, and parabolic cylinders.
3. Answer the following True and False questions; justifying your answers in words.
a) Two lines parallel to a third line are parallel.
True because for two lines to be parallel, their slopes have to be the equal. If two lines have the same slope as a third line, then the those two lines have the same slope as each other making them parallel.
b) Two lines perpendicular to a plane are parallel.
True because for a line to be perpendicular to a plane it has to have a specific slope and so all lines perpendicular to the plane will have the same slope making them all parallel.
c) Two lines perpendicular to a third line are parallel.
False because this statement is only true in two dimensions. When dealing with more dimensions, there is an infinite number of lines that are perpendicular to a line, none of which have to be parallel. These lines can be parallel, intersect, or be skew.
d) Two planes parallel to a third plane are parallel.
True for the same reason as parallel lines. For planes to be parallel, they have to be the same distance apart everywhere and never intersect. If this is true for two planes to a third plane. Then they can only be true for the two planes as well.
e) Two planes perpendicular to a line are parallel.
True because lines perpendicular to a plane are all the same slope so if two planes share the same perpendicular line, then the planes are also in the same direction and thus are parallel.
f) A plane and a line either intersect or are parallel.
True because a plane goes forever on all sides and a line goes forever in two directions. Thus the line can intersect the plane and the only time that the line does not is when the line is parallel.
g) Two planes either intersect or are parallel.
True because since planes go on forever on all sides, two planes will intersect every time except when they are parallel.
h) Two lines parallel to a plane are parallel.
False because if two lines are parallel to a plane, the line is equidistant to the plane at every point on the line however the line can be positioned to go in any direction as long as the line does not intersect the plane. So two lines parallel to a plane can be parallel, however they can also intersect or be skew.
i) Two lines intersect or are parallel.
False because in three-space, two lines can also be skew.
j) Two planes perpendicular to a third plane are parallel.
True because the planes perpendicular to one plane can only really be going in the same direction in different places along the third plane because planes go on forever on all sides.
k) Two planes parallel to a line are parallel.
True because if two planes are parallel to the same line they are both spread out in the same direction just in different places.