Math 312
Homework #1
Vector Functions and Space Curves
Show all work for full credit
1.
2.
Due Friday, September 15
Find a parametrization for the following:
a.
The line segment from the point 3, 1, 2 to the point 7, 1, 1.
b.
The curve of intersection of the surfaces z  y 2 and 2x  y  z  6 from
the point 3, 0, 0 to 0, 3, 9.
c.
The ellipse of intersection of the cylinder x 2  z 2  9 and the plane
x  2y  3z  1 oriented clockwise when looking from the positive y axis,
tracing the ellipse exactly once.
Consider the curve:
r t  sin t cos t i  sin 2 t j  cos t k
3.
Name:______________________________
Fall 2023
0  t  2
a.
Show algebraically that the curve lies on a sphere centered at 0, 0, 0
and state the radius of the sphere.
b.
At what point x, y, z in space does the tangent line to the curve at
t   intersect the xy  plane?
6
A fly is crawling along a wire helix so that its position vector function is:
r t  6 cost i  6 sint j  2t k for t  0.
a.
At what time will the fly hit the sphere x 2  y 2  z 2  100?
b.
At what point x, y, z in space will the fly be located at the time found in
part a?
c.
How far did the fly travel before it hit the sphere? Note: exact answer is
fine, it is not necessary to find a decimal approximation.
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