Which of the following vectorvalued functions represent the...

Which of the following vectorvalued functions represent the same graph? (a) $\mathbf{r}(t)=(-3 \cos t+1) \mathbf{i}+(5 \sin t+2) \mathbf{j}+4 \mathbf{k}$ (b) $\mathbf{r}(t)=4 \mathbf{i}+(-3 \cos t+1) \mathbf{j}+(5 \sin t+2) \mathbf{k}$ (c) $\mathbf{r}(t)=(3 \cos t-1) \mathbf{i}+(-5 \sin t-2) \mathbf{j}+4 \mathbf{k}$ (d) $\mathbf{r}(t)=(-3 \cos 2 t+1) \mathbf{i}+(5 \sin 2 t+2) \mathbf{j}+4 \mathbf{k}$

Which of the following vectorvalued functions represent the same graph?
(a) $\mathbf{r}(t)=(-3 \cos t+1) \mathbf{i}+(5 \sin t+2) \mathbf{j}+4 \mathbf{k}$
(b) $\mathbf{r}(t)=4 \mathbf{i}+(-3 \cos t+1) \mathbf{j}+(5 \sin t+2) \mathbf{k}$
(c) $\mathbf{r}(t)=(3 \cos t-1) \mathbf{i}+(-5 \sin t-2) \mathbf{j}+4 \mathbf{k}$
(d) $\mathbf{r}(t)=(-3 \cos 2 t+1) \mathbf{i}+(5 \sin 2 t+2) \mathbf{j}+4 \mathbf{k}$

Key Concepts

Graph Equivalence in Parameterizations

Graph equivalence in parameterizations refers to the concept that two different vector-valued functions can trace out the same geometric path even if they are expressed in different forms. By analyzing translations, reflections, and reparameterizations, one can determine whether seemingly different expressions correspond to the same graph, emphasizing the idea that the underlying geometry is independent of the specific parameterization used.

Trigonometric Parameterizations

Trigonometric parameterizations use sine and cosine functions to describe curves, often circles or ellipses. These functions naturally incorporate periodicity and oscillation, and adjustments such as amplitude changes, phase shifts, or frequency modifications typically affect the properties of the curve (such as size or orientation) but may still represent the same geometric shape under appropriate transformations.

Reparameterization

Reparameterization is the process of changing the parameter of a curve without altering the geometric shape of the curve. This means that different vector-valued functions, even if they use different parameter expressions or transformations (such as scaling the parameter or switching the roles of components), can represent the same set of points in space, thereby representing the same graph.

Parameterization of Curves

Parameterization of curves involves expressing the coordinates of points on a curve as functions of an independent parameter. This process gives us a way to describe the continuous set of points that form the curve and is particularly useful when dealing with curves that may not have an easily expressible Cartesian equation.

Vector-valued Functions

Vector-valued functions are functions where each input value is associated with a vector. In the context of curves, these functions map a parameter, often denoted by t, to a point in two- or three-dimensional space. This allows us to represent curves as well as more general geometric objects in a compact and useful way.

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