Problem 31 - Calculus Early Transcendentals

Question

(a) Show that the parametric equations $$x=x_{1}+\left(x_{2}-x_{1}\right) t \quad y=y_{1}+\left(y_{2}-y_{1}\right) t$$ where \(0 \leqslant t \leqslant 1,\) describe the line segment that joins the points \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right).\) (b) Find parametric equations to represent the line segment from \((-2,7)\) to \((3,-1)\).

Step-by-Step Solution

Answer

(a) The equations trace the line segment. (b) Parametric equations: \(x = -2 + 5t\), \(y = 7 - 8t\) for \(0 \leq t \leq 1\).

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x = x_{1} + (x_{2}-x_{1})t\) and \(y = y_{1} + (y_{2}-y_{1})t\) where \(0 \leq t \leq 1\). These equations are used to represent a line segment between two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\).

Step 2: Analyzing Endpoint Behavior

When \(t = 0\), we have \(x = x_{1}\) and \(y = y_{1}\), which means the parametric equations represent the point \((x_{1}, y_{1})\). When \(t = 1\), we have \(x = x_{2}\) and \(y = y_{2}\), representing the point \((x_{2}, y_{2})\). Thus, for \(0 \leq t \leq 1\), the equations trace the line segment from \((x_{1}, y_{1})\) to \((x_{2}, y_{2})\).

Step 3: Verifying Linearity

For any \(t\) between 0 and 1, the equations provide a linear interpolation between the endpoints, a direct consequence of adding a linear multiple of the difference between the endpoints to the initial point. This is typical for a line segment as all values \((x, y)\) lie in a straight line, confirming the shape is indeed a line segment.

Step 4: Defining Parametric Equations for Specific Points

Given the points \(P_1 = (-2, 7)\) and \(P_2 = (3, -1)\), we substitute these into the parametric equations. Calculate \( (x_2 - x_1) = 3 - (-2) = 5\) and \((y_2 - y_1) = -1 - 7 = -8\).

Step 5: Writing the Parametric Equations

Substitute the calculated values into the parametric equations: \(x = -2 + 5t\) and \(y = 7 - 8t\). These equations represent the line segment from \((-2, 7)\) to \((3, -1)\).

Key Concepts

Line Segment

A line segment is the part of a line that connects two endpoints. It is the straight path between two points in a space, and unlike a line, it has a definite beginning and end. When discussing parametric equations, a line segment is defined by the starting and ending points. Given two points, \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), their line segment can be described using parametric equations. These equations generate all the points in between \(P_1\) and \(P_2\) by varying a parameter, often denoted as \(t\). This parameter is constrained to values between 0 and 1.

• For \(t = 0\), the equations yield the starting point \(P_1\).
• For \(t = 1\), the equations yield the ending point \(P_2\).
• Any \(t\) between 0 and 1 provides points that lie on the line connecting \(P_1\) and \(P_2\).

This implies that line segments are a subset of lines, constrained by their endpoints.

Interpolation

Interpolation refers to a method of constructing new data points within the range of a discrete set of known data points. In the context of line segments and parametric equations, interpolation is the process of finding intermediate values between two endpoint values. The given parametric equations accomplish interpolation along the line segment:

• The variable \(t\) scales the difference between the endpoints.
• As \(t\) traverses from 0 to 1, each calculation in the equations gives a point on the segment between the two endpoints \(P_1\) and \(P_2\).
• This ensures that the generated line is smooth and linear, reflecting the characteristics of the segment.

Interpolation in this setting doesn't mean complex curves but rather a straightforward, linear interpolation often associated with line segments. In simpler terms, for any value of \(t\), these parametric formulas provide a precise point on the line segment, resulting in a evenly distributed set of points that form a straight path.

Endpoint Behavior

The behavior of parametric equations at endpoints is vital in verifying that they correspond with our intended line segment. The values that \(t\) can take, specifically 0 and 1, are crucial in identifying the segment's boundaries. As explained:

• When \(t = 0\), you retrieve the coordinates \((x_1, y_1)\), indicating that this is the starting point of the segment.
• When \(t = 1\), the coordinates transform into \((x_2, y_2)\), marking the segment's endpoint.
• For any \(t\) between (but not inclusive of) 0 and 1, the equations calculate points directly between these two coordinate extremes.

Endpoint behavior ensures that the entirety of the segment is described without going beyond its bounds. Each \(t\) maps precisely to a location on the line segment, ensuring a precise interpolation over this finite span.