Question

Parametrizing Lines and Segments (a) Show that \(x=x_{1}+\left(x_{2}-x_{1}\right) t, \quad y=y_{1}+\left(y_{2}-y_{1}\right) t\) \(-\infty

Short answer

The given equations indeed parametrize the line passing through the points \((x_1,y_1)\) and \((x_2,y_2)\) for all values of \(t => -\infty < t < \infty\). For the line segment specifically between these points, we use the same parametric equations, but restrict \(t\) to the interval \([0, 1]\).

Step-by-step solution

Understanding parametric representation of a line

A parametric equation represents the set of points on the plane satisfying the equation, with the parameter \(t\). In this case, you have two equations, one for \(x\) and one for \(y\), with \(t\) being the parameter. By varying \(t\) from \(-\infty\) to \(+\infty\), we can obtain every point on the line that passes through points \(P_1(x_{1}, y_{1})\) and \(P_2(x_{2},y_{2})\).

Verifying the parametric representation

By plugging \(t = 0\) into the parametric equations, it will demonstrate that the line passes through the point \(P_1(x_{1}, y_{1})\). Similarly, plugging \(t = 1\) will show that it also passes through the point \(P_2(x_{2}, y_{2})\).

Parametrizing a line segment

The parametrization of a line segment from \(P_1(x_{1}, y_{1})\) to \(P_2(x_{2},y_{2})\) will be the same parametric equations as in step 2, but this time the parameter \(t\) lies in the interval \([0,1]\), not \(-\infty<t<\infty\). This is because the line segment only includes the points between \(P_1\) and \(P_2\).

Key concepts

Parametrizing Line Segments

When it comes to understanding geometric concepts, parametrization plays a significant role. It refers to representing geometric objects, such as line segments, by defining a parameter that can take on a range of values. For line segments, we often use the parameter t as a way to express the coordinates of any point between two fixed endpoints.

Imagine you are walking from one point to another along a straight path. As you walk, your position can be described by how far along the path you've reached, which can be represented by t. When t=0, you're at the starting point, and when t=1, you've reached the end. This is essentially what we are doing when we parametrize a line segment using t.

The formula provided in the exercise,
\[ x = x_1 + (x_2 - x_1) t, \quad y = y_1 + (y_2 - y_1) t \]
can describe any point on the line segment when t lies within the interval [0, 1]. If t is outside of this interval, the corresponding point lies outside the line segment that connects the endpoints \((x_1, y_1)\) and \((x_2, y_2)\). This constraint on t is crucial for accurately parametrizing line segments.

Parametric Representation

A parametric representation defines the coordinates of points on a geometric object using one or more parameters. For instance, lines in the plane are often represented using a single parameter, typically denoted by t. The parametric equations provide a system where each value of t corresponds to a specific point on the line.

In the context of the exercise given,
\[ x = x_1 + (x_2 - x_1) t, \quad y = y_1 + (y_2 - y_1) t \]
we see a parametric representation of a line through points \((x_1, y_1)\) and \((x_2, y_2)\). By changing the value of t, which can range from negative infinity to infinity, we can trace every single point on this line infinitely in both directions.

This representation is powerful because it allows us to depict motion and direction, imagine dynamic changes, and calculate intermediate states. Additionally, it simplifies the process of calculating intersections and can be extended to represent curves and surfaces in higher dimensions.

Parametrization for Lines

Parametrization for lines extends the concept of a parametric representation to include linear paths of infinite length. Unlike the parametrization for line segments, where the parameter t is restricted to a specific interval [0, 1], here t can take any real number value.

For a line that extends infinitely in both directions, we express its parametric equations without bounds on t, which means t can be chosen from the entire set of real numbers, from \(-\infty\) to \(+\infty\).

Key Points in Line Parametrization

• It offers an elegant method to describe a line in a direction-oriented manner.
• It avoids the limitation of undefined slopes that often complicate the slope-intercept form.
• Parametrization aligns well with vector concepts, where we can think of the parametric equations as a 'base point' plus a 'direction vector' scaled by t.

By allowing the parameter to range freely, every point on the line is reachable and can be located by a specific value of t, making it invaluable for analyzing various mathematical, physical, and engineering problems where the concept of infinite or unbounded lines is relevant.