1C: Vectors and Matrices - Parametric Equations for Curves

Parametric equations define trajectories in space or in the plane. Very often we can think of the trajectory as that of a particle moving through space and the parameter as time. In this case, the parametric curve is written \((x(t); y(t); z(t))\), which gives the position of the particle at time t.

A moving particle also has a velocity and acceleration. These are vectors which vary in time. We will learn to compute them as derivatives of the position vector.

Last Modification: 2020-09-17 Thu 12:07

Captured On: [2020-01-18 Sat]
Source: Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare

Session 15: Equations of Lines

Captured On [2020-02-05 Wed 21:25] Source Session 15: Equations of Lines | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Equations of lines Figure 2: Example of line through 2 points Figure 3: Position at time \(t\) Figure 4: Get the parametric lines through 2 points 2 How does work a parametric curve? 2.1 Front How does work a parametric curve?...

Session 16: Intersection of a Line and a Plane

Captured On [2020-02-05 Wed 21:25] Source Session 16: Intersection of a Line and a Plane | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Application: Intersection with a plane Figure 2: Where are the points respect to the plane? Figure 3: When will be an intersection to the plane? Figure 4: What does happen when the line is parallel to the plane or in the plane...

Session 17: General Parametric Equations; the Cycloid

Captured On [2020-02-05 Wed 21:25] Source Session 17: General Parametric Equations; the Cycloid | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Parametric equations for arbitrary motion (plane/space) Figure 2: Cycloid path movement Figure 3: Which is the position \((x(\theta), y (\theta))\) Figure 4: Vector position \(\vec{OP}\) Figure 5: Getting vectors Figure 6: Vector position \(\vec{OP}\)...

Session 18: Point (Cusp) on Cycloid

Captured On [2020-02-05 Wed 21:26] Source Session 18: Point (Cusp) on Cycloid | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: What happens near bottom? Figure 2: Get an approximation for \(\theta\) small Figure 3: Taylor approximations for \(\sin\) and \(\cos\) when \(\theta\) is small Figure 4: Slope when \(\theta\) is small 2 How can we analyze what happens at cusps on a cycloid graph?...

Session 19: Velocity and Acceleration

Captured On [2020-02-05 Wed 21:26] Source Session 19: Velocity and Acceleration | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review parametric equations and position vector Figure 2: Velocity in a cycloid Figure 3: Calculating module of speed Figure 4: Acceleration a vector, and warning about derivative of a module 2 How can we get the velocity vector from a position vector?...

Session 20: Velocity and Arc Length

Captured On [2020-02-05 Wed 21:28] Source Session 20: Velocity and Arc Length | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Arc lenght Figure 2: Lenght of an arch of cycloid Figure 3: Velocity vector Figure 4: Limit as \(\Delta t \to 0\) 2 What does means speed? 2.1 Front What does means speed?...

Session 21: Kepler's Second Law

Captured On [2020-02-05 Wed 21:29] Source Session 21: Kepler’s Second Law | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Description of kepler’s second law Figure 2: Kepler’s law in terms of vectors? Figure 3: Area for second law Figure 4: Motion of the plane in the same plane Figure 5: Implications of Kepler’s second law...