Question

In Exercises \(23-28,\) find a parametrization for the curve. the line segment with endpoints \((-1,-3)\) and \((4,1)\)

Short answer

The parametrization of the line segment with endpoints \((-1,-3)\) and \((4,1)\) is \(P(t) = (1 + 3t, -3 + 4t)\), for \(0 \leq t \leq 1\).

Step-by-step solution

Identify the Points

Note the starting point \(P_0\) and ending point \(P_1\) as given in the question, which are \((-1, -3)\) and \((4, 1)\) respectively.

Apply the Parametrization Form to find X coordinate

For the x-coordinate, apply the parametrization form as follows: \(x(t) = (1 - t)*x_0 + t*x_1\). Here \(x_0\) is the x-coordinate of the starting point which is -1 and \(x_1\) is the x-coordinate of the end point which is 4. Substituting the known values, we get \(x(t) = (1 - t)*(-1) + t*4 = 1 - t + 4t = 1 + 3t\).

Apply the Parametrization Form to find Y coordinate

For the y-coordinate, similarly apply the parametrization form as follows: \(y(t) = (1 - t)*y_0 + t*y_1\). Here \(y_0\) is the y-coordinate of the starting point which is -3 and \(y_1\) is the y-coordinate of the end point which is 1. Substituting the known values, we get \(y(t) = (1 - t)*(-3) + t*1 = -3 + 3t + t = -3 + 4t\).

The Parametrization of the Line Segment

Combine the x and y coordinate functions to get the parametrization of the line segment. That is, \(P(t) = (x(t), y(t)) = (1 + 3t, -3 + 4t)\), for \(0 \leq t \leq 1\). This is the required parametrization of the given line segment.

Key concepts

Parametric Equations

Parametric equations represent a set of related quantities as functions of an independent variable, known aptly as the parameter. In coordinate geometry, particularly in the context of describing curves, these equations provide a way to express the coordinates of points on the curve as a function of one or more parameters. For instance, a line segment can be described using two parametric equations—one for the x-coordinate and one for the y-coordinate. Each equation takes a parameter, typically denoted as 't', which varies within a certain range.

Using the endpoint coordinates, the parameter 't' interpolates between these points. For the line segment with endpoints \( (-1,-3) \) and \( (4,1) \) as given in the exercise, the parameter 't' runs from 0 to 1. As 't' changes, the values of x and y change accordingly, tracing the path from the start point to the end point of the line segment. The final parametric equations for this segment are \( x(t) = 1 + 3t \) for the x-coordinate and \( y(t) = -3 + 4t \) for the y-coordinate.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This field of mathematics enables us to describe geometric shapes algebraically and make calculations with them using algebraic methods. By employing the Cartesian coordinate system, points on a plane are defined by a pair of numerical coordinates, which are distances to two or three perpendicular lines, known as axes.

In the context of our exercise, the points \( (-1,-3) \) and \( (4,1) \) are the coordinates of the endpoints of a line segment in a two-dimensional space. By linking these points with a straight line and assigning a parametric variable 't' to indicate the position on the line segment, every point can be precisely located using the corresponding x and y values computed from parametric equations.

Vector Algebra

Vector algebra is a branch of mathematics that deals with quantities that have both magnitude and direction. In the context of parametrizing a line segment, vectors are pivotal. The position vectors of the endpoints provide a starting point and direction. Parametric equations essentially blend these vectors according to the parameter 't'.

The vector leading from the origin to point \( P_0 = (-1,-3) \) is \( \vec{OP_0} \) and the vector leading to \( P_1 = (4,1) \) is \( \vec{OP_1} \). The vector from \( P_0 \) to \( P_1 \) is given by \( \vec{P_0P_1} = \vec{OP_1} - \vec{OP_0} \), which acts as the direction vector for the line segment. Parametric equations describe points along this line segment as a linear combination of this direction vector and the original position \( \vec{OP_0} \) adjusted by 't'.

Calculus

Calculus is a branch of mathematics focused on changes and motion. While the parametrization of a line segment doesn't immediately call to mind the complexities of calculus, the concepts of limits and the continuous nature of functions underpin the understanding of parametric equations.

For example, as 't' varies from 0 to 1 in our parametric equations, we're essentially mapping a continuous range of values onto a straight line, which is a fundamental idea in calculus—particularly integral calculus, where sums and areas under curves are considered. When you think about moving along the line segment over time, understanding calculus can help visualize the continuous movement described by the parametric equations. Moreover, in more advanced applications, derivatives of parametric equations are used to determine slopes, and integrals can find arc lengths, again showing the deep connection of calculus to coordinate geometry and vector algebra.