Vector-Valued Functions & Curve Parameterization

15 Q&A flashcards summarizing definitions, examples, and key properties of vector-valued functions, parameterizations of circles, helices, and intersections of surfaces.

What is the general form of a vector-valued function in ℝ³?

r(t) = (x(t), y(t), z(t)) where x(t), y(t), and z(t) are scalar component functions and t is a real-valued parameter.

In a vector-valued function r(t), what does the parameter t typically represent in physical problems?

Time.

If r(t) = (x(t), y(t), z(t)), what geometric object is traced out by the set of points (x(t), y(t), z(t))?

A curve (or path) in ℝ³.

How do you eliminate the parameter t to find an explicit Cartesian equation from r(t) = (t, t²)?

Solve for t from x = t (so t = x) and substitute into y = t² to get y = x².

What curve is represented by r(t) = (cos t, sin t, 1) for all real t?

A circle of radius 1 lying in the plane z = 1.

For r(t) = (3 cos t, 3 sin t, 4t), what is the projection of the curve onto the xy-plane?

A circle x² + y² = 9 of radius 3.

Why is r(t) = (3 cos t, 3 sin t, 4t) called a helix?

Because it wraps around the cylinder x² + y² = 9 while the z-coordinate increases linearly, creating a spiral (helical) path upward.

Give a convenient trig parameterization for the intersection of x² + y² = 25 and z = 2xy with y ≤ 0.

r(t) = (5 cos t, 5 sin t, 25 sin 2t) with π ≤ t ≤ 2π (so y = 5 sin t ≤ 0).

What radius and center correspond to the parameterization r(t) = (1, −1 + 2 cos t, 2 + 2 sin t)?

Radius 2, center (1, −1, 2).

Which coordinate remains constant for the circle r(t) = (3 sin t, 4, 3 cos t + 2)?

The y-coordinate; y = 4 for all t.

Determine the center and radius of r(t) = (3 sin t, 4, 3 cos t + 2).

Center (0, 4, 2) and radius 3.

How can you recognize that a vector expression such as x = 5 cos t, y = 5 sin t parameterizes a circle?

Because cos² t + sin² t = 1, giving x² + y² = 25, the equation of a circle of radius 5 centered at the origin in the xy-plane.

What does it mean to ‘parameterize a curve’?

To describe the curve’s coordinates as functions of a single variable (parameter) t.

In the context of vector-valued functions, what is the difference between scalar and vector functions?

A scalar function outputs a single real number, whereas a vector function outputs an ordered tuple (vector) of real numbers.

When the notes say a curve is ‘constrained to the plane z = 1,’ what does that imply about z(t)?

z(t) is constantly equal to 1 for all t.