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Parametric equations are a way to represent curves by using a parameter, usually denoted as \( t \). These equations allow you to express coordinates in terms of this single parameter, which makes them powerful for describing motion and paths in space. In the vector function \( \mathbf{r}(t) = 4 \mathbf{i} + 2\cos{t} \mathbf{j} + 3\sin{t} \mathbf{k} \), each component (\( x, y, \) and \( z \)) is expressed as a function of \( t \).

• X-component: The constant 4 represents a fixed position along the \( x \)-axis.
• Y-component: \( 2\cos{t} \) varies with \( t \) and describes oscillation along the \( y \)-axis.
• Z-component: \( 3\sin{t} \) varies with \( t \) and describes oscillation along the \( z \)-axis.

By varying \( t \), the curve traced is not just a static shape but a path, showcasing how all components relate over the same parameter.

A 3D coordinate system is used to provide a framework for locating any point within three-dimensional space, defined by three perpendicular axes: \( x, y, \) and \( z \). Each point can be uniquely identified by an ordered triplet \((x, y, z)\), where each value corresponds to a distance along an axis.In the given vector function problem, the 3D coordinate system is crucial because it allows us to visualize and graph multi-dimensional shapes, like the ellipse described by the function. Think of:

• X-Axis: Represents the direction in which the ellipse is shifted, held constant at its position of 4.
• Y-Axis and Z-Axis: The plane where the ellipse forms, defined by the parameters specific to \(2\cos{t}\) and \(3\sin{t}\).

To effectively interpret such equations, it's essential to understand how these axes interrelate and how transformations like shifts or rotations are applied in a 3D space. This understanding makes it simpler to plot curves and shapes as imparted by vector equations.

Ellipses are elongated circles, and in three-dimensional space, they can be oriented along any plane. The exercise demonstrates how ellipses are formed using parametric equations for the \( y \)- and \( z \)-components: \( 2\cos{t} \) and \( 3\sin{t} \). These describe an ellipse because they follow the general ellipse equation format: \( \frac{y^2}{a^2} + \frac{z^2}{b^2} = 1 \) when simplified.In our problem:

• Semi-Major Axis: Along the \( z \)-axis with length 3, because the coefficient of the \( \sin{t} \) term is 3.
• Semi-Minor Axis: Along the \( y \)-axis with length 2, since the coefficient of the \( \cos{t} \) term is 2.

The ellipse itself lies in the plane where the constant \( x \)-component is 4. Typically, the center of such an ellipse would be at the origin in 2D; however, due to the constant \( x = 4 \), it is shifted parallel to the \( yz \)-plane at this x-coordinate. Understanding this elliptical path's formation adds insight into motion and geometry within space.