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.
Last Modification: 2020-09-17 Thu 11:44
Captured On [2020-02-05 Wed 19:40] Source Session 1: Vectors | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Definition of vectors Figure 2: Length of a vector Figure 3: Modules and addition Figure 4: Multiplying by scalars 2 Which is the vector between 2 points? 2.1 Front Which is the vector between 2 points?...
Captured On [2020-02-05 Wed 19:41] Source Session 2: Dot Products | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Dot Product Definition Figure 2: What does geometic definition mean? Figure 3: Dot product of combined vectors 2 What is the dot product of 2 vectors? 2.1 Front What is the dot product of 2 vectors?...
Captured On [2020-02-05 Wed 19:43] Source Session 3: Uses of the Dot Product: Lengths and Angles | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Computing lengths and angles Figure 2: Resolution of computing an angle Figure 3: Meaning of sign of a dot product Figure 4: Detect orthogonality Figure 5: Plane throught \(O\), perpendicular to \(\vec{A}\)
Captured On [2020-02-05 Wed 19:46] Source Session 4: Vector Components | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review of dot product Figure 2: Components of \(\vec{A}\) along direction \(\vec{u}\) Figure 3: Pendulum problem with projections 2 Which is it the angle between 2 vectors? 2.1 Front Which is it the angle between 2 vectors?...
Captured On [2020-02-05 Wed 19:47] Source Session 5: Area and Determinants in 2D | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area within 2 vectors Figure 2: Rotated vector \(\vec{A}\) Figure 3: Area as determinant of 2 vectors Figure 4: Area of parallelogram and triangle 2 What is the area between 2 vectors? 2....
Captured On [2020-02-05 Wed 19:49] Source Session 6: Volumes and Determinants in Space | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Determinant in space Figure 2: Volume of parallelepiped 2 What is the volume between 3 vectors? 2.1 Front What is the volume between 3 vectors? 2.2 Back With determinants \(\abs{\det(\vb{A}, \vb{B}, \vb{C})}\) With cross product Volume = area(base) height \(V = \abs{\vb{A} \cross \vb{B}} (\vb{C} \cdot \hat{n})\) \({\displaystyle \hat{n} = \frac{\vb{A} \cross \vb{B}}{\abs{\vb{A} \cross \vb{B}}}}\) \({\displaystyle V = \abs{\vb{A} \cross \vb{B}} (\vb{C} \cdot \frac{\vb{A} \cross \vb{B}}{\abs{\vb{A} \cross \vb{B}}}) = \vb{C} \cdot (\vb{A} \cross \vb{B})}\) 3 What happens if \(\vb{a} \cross \vb{b} = 0\)?...
Captured On [2020-02-05 Wed 19:50] Source Session 7: Cross Products | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Cross product of 2 vectors in 3-space Figure 2: Area of parallelogram with cross product Figure 3: Right hand method for direction of cross product Figure 4: Another look at volume Figure 5: Volume of parallelopiped...
Captured On [2020-02-05 Wed 19:52] Source Session 8: Equations of Planes | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Equation of plane Figure 2: Another solution equation of plane 2 How can we find the equation of the plane containing 3 points? 2.1 Front How can we find the equation of the plane containing 3 points?...