1. Vectors and Matrices

This unit covers the basic concepts and language we will use throughout the course. Just like every other topic we cover, we can view vectors and matrices algebraically and geometrically. It is important that you learn both viewpoints and the relationship between them.

Part A: Vectors, Determinants, and Planes

» Session 1: Vectors
» Session 2: Dot Products
» Session 3: Uses of the Dot Product: Lengths and Angles
» Session 4: Vector Components
» Session 5: Area and Determinants in 2D
» Session 6: Volumes and Determinants in Space
» Session 7: Cross Products
» Session 8: Equations of Planes
» Problem Set 1

Part B: Matrices and Systems of Equations

» Session 9: Matrix Multiplication
» Session 10: Meaning of Matrix Multiplication
» Session 11: Matrix Inverses
» Session 12: Equations of Planes II
» Session 13: Linear Systems and Planes
» Session 14: Solutions to Square Systems
» Problem Set 2

Part C: Parametric Equations for Curves

» Session 15: Equations of Lines
» Session 16: Intersection of a Line and a Plane
» Session 17: General Parametric Equations; the Cycloid
» Session 18: Point (Cusp) on Cycloid
» Session 19: Velocity and Acceleration
» Session 20: Velocity and Arc Length
» Session 21: Kepler’s Second Law
» Problem Set 3

Exam 1

» Practice Exam
» Session 22: Review of Topics
» Session 23: Review of Problems
» Exam

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Vectors are basic to this course. We will learn to manipulate them algebraically and geometrically. They will help us simplify the statements of problems and theorems and to find solutions and proofs.

Determinants measure volumes and areas. They will also be important in part B when we use matrices to solve systems of equations.

» Session 1: Vectors
» Session 2: Dot Products
» Session 3: Uses of the Dot Product: Lengths and Angles
» Session 4: Vector Components
» Session 5: Area and Determinants in 2D
» Session 6: Volumes and Determinants in Space
» Session 7: Cross Products
» Session 8: Equations of Planes
» Problem Set 1

The basic point of this part is to formulate systems of linear equations in terms of matrices. We can then view them as analogous to an equation like 7_x_ = 5.

In order to use them in systems of equations we will need to learn the algebra of matrices; in particular, how to multiply them and how to find their inverses.

Geometrically, a linear equation in x, y and z is the equation of a plane. Solving a system of linear equations is equivalent to finding the intersection of the corresponding planes.

» Session 9: Matrix Multiplication
» Session 10: Meaning of Matrix Multiplication
» Session 11: Matrix Inverses
» Session 12: Equations of Planes II
» Session 13: Linear Systems and Planes
» Session 14: Solutions to Square Systems
» Problem Set 2

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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.

» Session 15: Equations of Lines
» Session 16: Intersection of a Line and a Plane
» Session 17: General Parametric Equations; the Cycloid
» Session 18: Point (Cusp) on Cycloid
» Session 19: Velocity and Acceleration
» Session 20: Velocity and Arc Length
» Session 21: Kepler’s Second Law
» Problem Set 3

« Previous | Next »

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Multivariable Calculus