Answer
Elliptical helix
Work Step by Step
Step 1 In this exercise, they want us to describe the curve represented by the vector-valued function: \[ \mathbf{r} = a\cos t \mathbf{i} + b\sin t \mathbf{j} + ct \mathbf{k} \] where $a$, $b$, $c$ are positive constants and $a \neq b$. Step 2 This combination of components represents an ellipse when $a \neq b$ (and a circle when $a = b$): \[ a\cos t \mathbf{i} + b\sin t \mathbf{j} \] One full ellipse is traced when $t$ increases by $2\pi$. Step 3 The $\mathbf{i}$ and $\mathbf{j}$ components of $\mathbf{r}$ describe an ellipse for the $x$ and $y$ parts of the curve. Now we consider the effect of the $\mathbf{k}$ component, which is $\mathbf{z} = ct$. $c$ is a constant, so $\mathbf{z}$ increases linearly as $t$ increases. So the curve is tracing out ellipses while moving in the positive $\mathbf{z}$ direction. It is an elliptical helix that is going around an elliptic cylinder with the $\mathbf{z}$-axis as its axis. This is similar to the earlier exercise with the conical helix, but the lengths of the minor and major axes of the ellipses are not changing since $a$ and $b$ are constant. Step 4 By recognizing the curve represented by certain combinations of vector-valued function components, then considering the function as a whole, we were able to understand that the curve represented an elliptical helix. Result The curve is an elliptical helix.