Final answer:
A vector function to represent the curve of intersection of a cylinder is expressed using parameters to describe the curve in a coordinate system with the origin at the cylinder's center. For a helical curve, it can be in the form of r(t) = , where R is the radius, θ(t) is the angular parameter, and h(t) is the height above the base.
Step-by-step explanation:
To find a vector function representing the curve of intersection of a cylinder, we need to use parameters to describe the curve in three-dimensional space. The given terms seem to relate to the motion of a particle in a circular path within a coordinate system, where the origin is the cylinder's center.
Assuming we're dealing with a cylinder aligned with the z-axis, we can express the position vector as r(t) = <R cos(θ(t)), R sin(θ(t)), h(t)>. Here, R would be the radius of the cylinder, θ(t) is the angle parameter, which may be a function of time, and h(t) describes the height of the point on the cylinder as a function of time.
Given that we have an example r(t) = <2t, t, sin(t)>, we can assume this is the position vector for some curve, possibly a helix on the cylinder's surface. The coefficients of t in the x and y components suggest a linear relationship with time, while the sin(t) component in z suggests a harmonic vertical motion, which is typical for a helix on the surface of a cylinder.