Lec 5: Parametric equations for lines and curves | MIT 18.02 Multivariable Calculus, Fall 2007

The script begins with a review of 3x3 linear systems and their geometric interpretation in terms of intersecting planes. It explains scenarios where a unique solution may not exist, such as when the intersection line of two planes is parallel to the third plane, leading to either no solution or infinitely many solutions. The lecture then transitions into discussing the equations of lines, initially describing them as intersections of two planes but quickly shifting to the parametric representation of lines as a more convenient approach. The concept of a line as a trajectory of a moving point is introduced, setting the stage for the exploration of parametric equations of lines.

This paragraph delves into the specifics of parametric equations of lines, illustrating how to derive them using two points on the line or a point and a directional vector. It provides an example using two points in space, (-1,2,2) and (1,3,-1), to demonstrate the process of finding the parametric equations. The explanation includes the concept of a moving point on the line, described by its position at various times, and the method to find the position of this point at any given time t. The paragraph concludes with the derivation of the parametric equations for the line defined by the two example points.

The script continues with an application of parametric equations to determine the relative positions of a line and a plane. It introduces a plane with the equation x + 2y + 4z = 7 and explores whether a given line intersects this plane and, if so, where. The discussion involves evaluating the plane equation at two points on the line to determine if they are on the same side, opposite sides, or if one lies on the plane. The paragraph also includes an interactive element where the audience is asked to vote on the relative positions, followed by a clarification of the correct answer based on the calculations.

Building on the previous discussion, this paragraph focuses on finding the exact point of intersection between a line and a plane using parametric equations. It guides through the process of substituting the parametric equations of the line into the plane's equation and solving for the parameter t that satisfies the plane equation. The example provided finds that the line intersects the plane at t = 1/2, and the corresponding point of intersection is calculated. The paragraph also addresses what happens if no solution exists, indicating that the line would be parallel to or contained within the plane.

The script broadens the application of parametric equations to model not just lines, but any curve or trajectory of a moving point. It introduces the concept with an example of a cycloid, a curve traced by a point on the rim of a rolling wheel. The paragraph explains the physical setup of a wheel rolling on a horizontal surface and the trajectory of a point on the wheel's circumference. It sets the stage for deriving the parametric equations for the cycloid in subsequent paragraphs.

This paragraph continues the exploration of cycloids by attempting to derive their parametric equations. It introduces the parameters of the problem, such as the radius of the wheel and the angle of rotation, and discusses the relationship between the distance traveled by the wheel and the arc length on the circumference. The explanation involves breaking down the position vector of the point on the wheel into simpler vectors, which are then used to express the parametric equations for the cycloid's curve.

The script examines the behavior of the cycloid near its bottom point, a feature that appears pointy in graphical representations. It discusses the possibility of the curve having a corner, a smooth transition, or even a loop at that point. The paragraph uses the derived parametric equations and simplifies them by rescaling with the wheel's radius as the unit of length. It then applies Taylor expansions to approximate the sine and cosine functions for small angles, providing insight into the curve's behavior near the bottom point.

This paragraph concludes the analysis of the cycloid's behavior near the origin by calculating the ratio of the y-component to the x-component of the parametric equations. It shows that as the angle approaches zero, the slope of the curve becomes infinite, indicating a vertical tangent at the origin. The explanation clarifies that the dominant motion near the origin is vertical, confirming the presence of a cusp, or a pointy feature, in the curve's geometry.

The script concludes with an announcement about the end of the lecture and the plan for the upcoming week. It informs the audience that practice exams will be provided for the following week's test, which will take place on Tuesday. The paragraph serves as a reminder and a transition to the next part of the course.