**Vectors and The Geometry of Space**

## Three-Dimensional Coordinate Systems

54 min 10 Examples

- Introduction to the 3D Coordinate System and the Right Hand Rule
- How do planes divide space?
- Discovering the 8 Octants and Learning how to plot points in 3-Space
- Set Notation Overview
- Graphing Planes in 3-Space (2 examples)
- Graphing a Circle and Cylinder in 3-Space
- Distance and Midpoint Formulas for 3-Space
- Discovering the Equation for a Sphere
- Example #1 of writing an equation of a sphere
- Example #2 of writing an equation of a sphere
- Example #3 of writing an equation of a sphere by completing the square

## Vectors in 3D

28 min 4 Examples

- Overview and Notation for Vectors in 2D and 3D
- Understanding vector addition and subtraction
- Overview of Vector Properties
- Discovering the Unit Vector
- Example of Writing Vectors in 3-Space
- Example of an Algebraic Proof for Associative Property for Vectors

## Dot Product in 3D

31 min 5 Examples

- Dot Product Definition and Example
- Example of finding the cosine of the angle between two vectors
- Example of how to determine if two vectors are orthogonal
- Directional Angles and Directional Cosines
- Example of finding Directional Cosines
- Example of finding a vector given Directional Angles
- Overview of Vector Projection
- Example of finding Scalar and Vector Projection

## Cross Product in 3D

41 min 5 Examples

- Cross Product Definition
- What is a Matrix and how to we find a Determinant for a 2×2 Matrix?
- Formula for the Cross Product using Expansion by Minors for a 3×3 Determinant
- Example of finding the Cross Product of two vectors
- Overview of how to find the Area of a Parallelogram and Determining if two vectors are parallel
- Example of finding the Area of a Parallelogram
- Overview of calculating the volume of a Parallelepiped
- Example of finding the volume of a parallelepiped
- Summary of all Dot Product and Cross Product Properties

## Equations of Lines and Planes

1 hr 13 min 8 Examples

- Overview of how to Write Equations of Lines in 3-Space
- Writing Equations of Line in Vector, Parametric, and Symmetric Form
- Example #1 of writing an equation of a line in 3-space
- Example #2 of writing an equation of a line in 3-space
- Example of how to determine if a line intersects or is parallel to a plane
- Example of how to determine if two lines are parallel, perpendicular, intersecting or skew
- Overview of how to Write Equations of Planes
- Example #1 of Writing and Graphing an Equation of a Plane
- Example #2 of Writing an Equation of a Plane
- Determining if two planes are Parallel or Perpendicular
- How to calculate the distance from a point to a plane

## Cylinders and Quadric Surfaces

49 min 9 Examples

- Overview of 2D Conic Sections
- Identifying Cylinders
- Identifying Ellipsoids
- Identifying Cones
- Identifying Hyperboloids
- Identifying Paraboloids
- Finding Domain for 3D Surfaces
- Three additional Examples of finding Domain
- Overview of Level (Contour) Curves with Example
- Example of Drawing Level Curves
- Overview of Finding Traces with 3 Examples
- Example of Sketching Traces

## Cylindrical and Spherical Coordinates

1 hr 9 min 16 Examples

- Review of Polar Coordinates and Introduction to Cylindrical Coordinates
- Two Examples of converting from Rectangular to Cylindrical Coordinates
- Two Examples of converting from Cylindrical to Rectangular Coordinates
- Two Examples of converting Functions from Cylindrical to Rectangular
- Three Examples of converting Functions from Rectangular to Cylindrical
- Overview of the Spherical Coordinate System
- Example #1 of converting from Spherical to Rectangular Coordinates
- Example #2 of converting from Spherical to Rectangular Coordinates
- Example #1 of converting a function from Spherical to Rectangular
- Example #2 of converting a function from Spherical to Rectangular
- Example of converting from Rectangular to Spherical Coordinates
- Example #1 of converting a function from Rectangular to Spherical
- Example #2 of converting a function from Rectangular to Spherical
- Summary of Cylindrical and Spherical Coordinates