Problem 8 - Advanced Engineering Mathematics

Question

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=t \mathbf{i}+t^{3} \mathbf{j}+t \mathbf{k} $$

Step-by-Step Solution

Answer

The graph is a twisted cubic curve lying in the plane where \( x = z \) and \( y = x^3 \).

Step 1: Analyze the Vector Function

The given vector function is \( \mathbf{r}(t) = t \mathbf{i} + t^3 \mathbf{j} + t \mathbf{k} \). This expression defines a path traced by varying the parameter \( t \). Each component describes the position of the point in the 3D Cartesian coordinate system: \( x = t \), \( y = t^3 \), and \( z = t \).

Step 2: Understand the Geometry

We have the following relationships from the vector components: \( x(t) = t \), \( y(t) = t^3 \), and \( z(t) = t \). Notice that both \( x \) and \( z \) are linearly related to \( t \). Thus, \( x = z \) which means we are constrained to the plane defined by \( x = z \).

Step 3: Eliminate the Parameter

To graph the curve, we need to eliminate \( t \) from the equations. Since \( x = t \) and \( z = t \), we have the simple relationship \( x = z \). For \( y \), we have \( y = t^3 \). Substitute \( t = x \) into \( y(t) \): Now \( y = x^3 \). The parametric equations are then reduced to \( y = x^3 \) in the plane \( x = z \).

Step 4: Plot the Curve in 3D

Plot the curve by considering the equations \( x = z \) and \( y = x^3 \). This means any point (x, y, z) on the curve satisfies both \( y = x^3 \) and \( x = z \). Therefore, the curve is a twisted cubic curve that lies in the plane \( x = z \), and extends through all space where \( x \), \( y \), and \( z \) satisfy these conditions.

Key Concepts

Parametric Equations

In mathematics, a parametric equation defines a group of quantities as explicit functions of some independent variables called parameters. Unlike traditional equations that express each dependent variable independently in terms of other variables, parametric equations introduce an additional variable that influences multiple results. Here, the vector function \( \mathbf{r}(t) = t \mathbf{i} + t^3 \mathbf{j} + t \mathbf{k} \) serves as a fine example.

• Each component of the vector function depends on the parameter \( t \).
• The functions \( x(t) = t \), \( y(t) = t^3 \), and \( z(t) = t \) individually describe how each coordinate changes with \( t \).

This joining of expressions allows the depiction of more complex curves and surfaces, providing an essential tool for describing paths in physics, animation, and other visualizations.

3D Cartesian Coordinates

Cartesian coordinates are a standard coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates. When extended to three dimensions, it allows us to locate points in space with three numbers, often denoted as \( (x, y, z) \). In the context of the exercise, these coordinates correspond to points on a curve defined by the vector function.

• The relationship \( x = t \), \( y = t^3 \), and \( z = t \) shows how our curve exists in a 3D space.
• This representation tells us the position of each point along the curve.

The 3D Cartesian system powerfully illustrates three-dimensional relationships and behaviors, underpinning much of geometry, computer graphics, and physics.

Graphing Curves

Graphing curves involves translating the parameters and relationships into a visual form. Given the parametric expressions, visualizing the curve means interpreting the geometry defined by these coordinates. In our exercise, the curve formed by the vector function \( \mathbf{r}(t) \) materializes in a specific 3D space, laying out a known path derived from the equations.

• Points on the curve meet the conditions \( y = x^3 \) and \( x = z \).
• The path can be identified as a twisted cubic, a common appearance for conditions of this nature.

Being able to visualize the behavior of the curve—as it charts among the constraints of \( x, y, \) and \( z \)—is a cornerstone of three-dimensional graphing, lending itself to advanced studies of geometry and modeling in applied fields like computer graphics and engineering.