MATH 260
EXAM 1
1.
(30
pt) Consider the points 
(1,2,-1), 
(2,2,0), 
(-3,1,-2).
a)
Compute
the vectors a=
and b=
.
b)
Compute
a
b and a
b.
c)
What
is the angle between a and b?
d)
Compute
the vector projection of a on b.
e)
Find
the line determined by 
and 
.
f)
Find
the plane determined by all three points.
2.
(10
pt) Classify and sketch a picture of the quadratic surface 
.
3.
(20
pt) At time t=0 a particle is at the origin and has velocity vector 
. At any time, its acceleration is given by a
.
a)
Find
the vector-valued function that gives the position of the particle.
b)
How
fast is the particle travelling when t=ln(2)?
4.
(10
pt) Find the point on the graph of 
where the curvature
is a maximum. (Justify that you have a maximum any way you want).
5.
(20
pt) Consider the set of all points in 3-space satisfying 
.
a)
Describe
this object is 3-space.
b)
Rewrite
the equation in terms of cylindrical and spherical coordinates.
6.
(10
pt) You travel along the vector valued function 
(
is measured in miles). When you come to the point (1,2,1),
you take off on the tangent vector and go for 5 miles. At what point in space
do you end up?
7.
(10
pt)(EXTRA CREDIT). Show that the distance from the point 
to the line 
is given by the
formula:
(Hint:
Scalar projection).