Problem 21 - Thomas Calculus

Question

Find a parametrization for the curve. the line segment with endpoints \((-1,-3)\) and \((4,1)\)

Step-by-Step Solution

Answer

The parametrization is \(x(t) = -1 + 5t\) and \(y(t) = -3 + 4t\), with \(0 \leq t \leq 1\).

Step 1: Identify the points

The line segment is between two points: the starting point \((-1, -3)\) and the ending point \(4, 1)\). These points will help us find a parameterization.

Step 2: Calculate the direction vector

Find the direction vector by subtracting the starting point from the ending point. \[\begin{align*}\text{Direction vector} &= (4, 1) - (-1, -3) \&= (4 - (-1), 1 - (-3)) \&= (5, 4).\end{align*}\]

Step 3: Parametrize the line segment

To parametrize the line segment, use the formula for a parameterized line: \[\mathbf{r}(t) = (1-t) \mathbf{a} + t \mathbf{b}, \]where \((x(t), y(t)) = (x_1 + t(x_2 - x_1), y_1 + t(y_2 - y_1))\) for \(0 \leq t \leq 1\). Here, \((x_1, y_1) = (-1, -3)\) and \((x_2, y_2) = (4, 1)\).

Step 4: Write the parametrized equation

Using the formula, the parametrization of the line segment is:\[\begin{align*}x(t) &= -1 + 5t, \y(t) &= -3 + 4t, \\end{align*}\]where \(0 \leq t \leq 1\).

Key Concepts

Parametric Equations

Parametric equations allow us to represent geometric shapes and curves in a way that reveals their characteristics more naturally. Instead of expressing the curve as a single equation in two variables like "y = f(x)", we use a parameter, usually "t", to describe both "x" and "y" as functions of "t". This approach is particularly useful for describing line segments, curves, and even surfaces in a 3D space.

Here, with the line segment between points \((-1, -3)\) and \((4, 1)\), parametric equations are perfectly suited because they prescribe both coordinates simultaneously using the same parameter. The benefits are:

• Each value of "t" corresponds to a unique point on the line.
• The equations are continuous and can represent finite segments effortlessly.

For our equation, "x(t) = -1 + 5t" and "y(t) = -3 + 4t" give us an easy way to compute any point on the segment as "t" varies from 0 to 1.

Line Segment

A line segment is a fundamental concept in geometry, defined as the part of a line that connects two points. Unlike a line, which extends indefinitely in both directions, a line segment has two endpoints.

In parametrizing a line segment, understanding these endpoint coordinates is crucial because they are used to define the start and end of the parameterized path. For example, with our endpoints \((-1, -3)\) and \((4, 1)\), we can navigate between them through varying the parameter "t" between 0 and 1.

• When "t = 0", the point corresponds to \((-1, -3)\).
• When "t = 1", the point corresponds to \(4, 1)\).
• Values of "t" between 0 and 1 give points along the segment.

The concept of a line segment ensures we're dealing with a finite piece of geometric space, providing a clear start and finish to our investigation.

Direction Vector

The direction vector is a crucial concept in vectors and geometry, as it signifies the direction and, implicitly, the distance between two points along a path.

For any line or segment, the direction vector can be constructed by subtracting coordinates of the start point from those of the end point. For our segment from \((-1, -3)\) to \(4, 1)\), this results in a direction vector of \(5, 4)\).

• This vector describes both the magnitude and direction in which to travel from one endpoint to the other.
• The vector components, 5 and 4, tell us to move 5 units along the x-axis and 4 units along the y-axis.

Using the direction vector, we can clearly write parametric equations that encode these movements, allowing the precise traversal from one endpoint to the other.