1. (15 points) Let H1be the plane x+ 2yโ2z= 1 and H2the plane
y= 2. Describe by equation and in words the set of all points Psuch
that the distance of Pfrom H1is the same as the distance of Pfrom
H2.
2. (15 points) Classify the following quadric surfaces and sketch their
graphs. (including: name the surfaces, indicate the axes and mark
the special points.)
(a) x2
โz2= 0,
(b) x2+ 2y2+ 4z2=x+ 2y+ 4z,
(c) x2+y2
โz= 1,
(d) 4x2+y2
โz2= 2yโ2z,
(e) x2+y2
โz2= 1.
3. (15 points) The path of a spaceship, which started long ago and far
away, is given by a vector valued function r(t) = (t, t2, t3
โt).
a) Find the times twhen the velocity is perpendicular to the accelera-
tion.
b) Draw and describe by equation the orthogonal projection of the path
onto the (x, y) plane.
b) Draw and describe by equation the orthogonal projection of the path
onto the (x, z) plane.
d) Suppose that at time t= 2 a bolt breaks off the spaceship and
afterwards moves along a straight line Lwith constant velocity. What
will then be the position of the bolt at time t= 5?
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