18.02, Spring 1999 -- Brief Lecture notes, First part

• Lecture 1: Feb 2.
  1. Vectors in 3 space, addition, scalar multiplication, length, unit vectors.
  2. Parallelogram formed from mid-points of sides of a quadrilateral.
  3. Force on a pendulum.
  4. Sclar product, components, length, orthogonality.
• Lecture 2: Feb 4.
  1. Scalar product formula proved.
  2. Area of planar parallelogram derived.
  3. Volume of parallelepiped given as determinant.
  4. Computation of 3 x 3 determinants.
  5. Vector product defined (geometrically and algebraically).
  6. Use of vector product.
• Lecture 3: Feb 5.
  1. Discussed length of vector product as area of parallelogram.
  2. Linear systems, especially 3 x 3 and 2 x 2.
  3. Matrices, acting on vectors.
  4. Matrix algebra -- product of matrices
  5. Solution of linear equation and invertibility of matrix.
  6. Computed inverse of a 3 x 3 matrix.
• Lecture 4: Feb 9.
  1. Inverse of 3x3 linear system again.
  2. Equation for a plane, plane with given normal through a point.
  3. Example: Plane through 3 points by cross product
  4. Cramer's rule -- indication of proof.
  5. Solutions of homogeneous 3x3 if determinant vanishes.
• Lecture 5: Feb 11.
  1. Parametric equations for a line.
  2. Example: Interseciton of line through (1,-2,0) and (-1,4,2) and the plane x+y+z=2.
  3. Example: Line formed by intersection of two planes.
  4. Parametric equations for curves, circle, cycloid.
  5. Derivative of a curve -- velocity.
• Lecture 6: Feb 12.
  1. Speed
  2. Acceleration
  3. Kepler's laws
  4. Central motion