Answer:
(a) r1(t) = <2cost, 0, 2sint>
(b) <2cost, (1 - 12sint - 10cost)/8, 2sint>
Step-by-step explanation:
x²+z²=4
a)
Within the xz plane, since y = 0...
Thus, x² + z² = 4 describes a circle centered at (0,0)..
This can be parameterized as
x = 2cos(t)
z = 2sin(t)
The necessary parameterization is:
r1(t) = <2cost, 0, 2sint>
b)
Cylinder equation: x² + z² = 4
Plane equation: 5x+8y+6z=1
Once again, the x² + z² = 4 can be parameterized as
x = 2cost, z = 2sint
Now, we can determine y using the plane equation...
5x+8y+6z=1
5(2cost) + 8y + 6(2sint) = 1
8y = 1 - 12sint - 10cost
y = (1 - 12sint - 10cost)/8
Thus, the parameterization becomes:
<2cost, (1 - 12sint - 10cost)/8, 2sint>