91Ó°ÊÓ

In mathematics, a parametric equation uses parameters to define a set of related mathematical equations. Here, each variable is expressed as a function of one or more independent parameters. This is particularly useful in describing curves in space or on a plane.
In the context of vector valued functions, we represent our curve through a parametric form that describes the position of a point in terms of a parameter, typically denoted by \( t \). For our given equation \( \mathbf{r} = -3 \mathbf{i} + (1 - t^2) \mathbf{j} + t \mathbf{k} \), each component can be understood as:

• \( x = -3 \): a constant value for the \( x \)-coordinate.
• \( y = 1 - t^2 \): a changing \( y \)-coordinate depending on \( t \).
• \( z = t \): the \( z \)-coordinate directly controlled by \( t \).

This parametric form allows us to visualize how a point moves through 3D space as \( t \) varies, producing what is known as a parametric curve. Such descriptions are fundamental in physics and engineering for describing motion and paths of objects.

3D space, short for three-dimensional space, refers to the geometric setting in which three values (coordinates) determine the position of an element, such as a point. This consists of three axes usually labeled as \( x \), \( y \), and \( z \).
The concept of 3D space is vital to understanding how objects and shapes are positioned and oriented in our physical world. For a vector valued function like \( \mathbf{r} = -3 \mathbf{i} + (1 - t^2) \mathbf{j} + t \mathbf{k} \), it shows us how a point's position changes in this space horizontally along the \( x \)-axis, vertically along the \( y \)-axis, and depth-wise along the \( z \)-axis.

• The \( x \)-coordinate stays constant, representing a vertical plane where \( x = -3 \).
• The \( y \) and \( z \) coordinates demonstrate changes as functions of the parameter \( t \).

Understanding 3D space is crucial to visualize and interpret the geometrical arrangement of points, lines, and shapes, especially when considering problems involving movement and trajectories.

A parabola is a symmetric curve defined by a quadratic function, commonly found in mathematics. When focusing on the \( yz \)-plane, we consider a 2D vertical plane parallel to both the \( y \)- and \( z \)-axes.
In the provided equation \( \mathbf{r} = -3 \mathbf{i} + (1 - t^2) \mathbf{j} + t \mathbf{k} \), observing the relationship between \( y \) and \( z \) helps describe the shape. Since \( y = 1 - z^2 \) when \( z = t \), we see a parabolic curve formed within the \( yz \)-plane. Important characteristics include:

• Vertex at the point \( (y, z) = (1, 0) \).
• Opens downwards, as indicated by the negative coefficient of \( z^2 \).
• Symmetry along the \( z \)-axis where \( z = 0 \).

This parabola is restricted to the plane \( x = -3 \), resulting in a cylindrical parabolic form in 3D space. Understanding such curves is useful in various fields, including physics, where parabolic paths describe projectile motion under uniform gravity, and computer graphics for rendering authentic trajectories and shapes.