Topic: Vectors and Geometry of Space

Subtopic: Rectangular Coordinates and Vectors in 3D

**Overview**

Giving our graphs some dimension, to day we expand our graphs from a 2D system to a 3D system with x-axis, y-axis, and z-axis. No longer are we constrained to four quadrants as now we can graph lines, curves, planes, and 3D shapes in all eight octants. Additionally, we look at vectors in 3D. All the operations and processes that we can do with vectors in a plane (2D vectors) we will now perform on vectors in space (3D vectors). Visually appealing fun stuff!

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 13.2.1 Know terminology related to a 3D axis system (eg. yz-plane) and understand what it means to be a right-handed system
- 13.2.2 In a 3D system plot points, find distance between two points, find the midpoint of the line segment spanning two points
- 13.2.3 Given the equation of a plane, sketch it on a 3D system
- 13.2.4 Given the sketch of a plane in 3D or given information about a plane in 3D, find its equation
- 13.2.5 Determine from the sketch and/or equations if two planes are parallel or not
- 13.2.6 Given the equation of a sphere or ball, sketch it on a 3D system
- 13.2.7 Given the sketch of a sphere or ball in 3D or given information about a sphere or ball in 3D, find its equation
- 13.2.8 Perform vector operations on vectors in 3D
- 13.2.9 Do all the things that we learned to do on 2D vectors in the plane now on 3D vectors in space including find magnitude, direction, and a unit vector in the direction of a given vector

**Terminology**

Define: three-dimensional rectangular coordinate system, right-handed systems, octant, ordered triple, sphere, ball, ℝ^{3}

Formulae to have in your notes: distance, midpoint

**Supplemental Resources (recommended)**

3D Vector Plotter from Acedemo is user-friendly free software to display 3D vectors, make adjustments to see the effect, and spin to view from different perspectives. Academo also has a limited but easy-to-use 3D Surface Plotter.

CalcPlot3D is a fairly comprehensive free interactive colour 3D graphing software (Java-based) worth checking out if you need one. Detailed directions at CalcPlot3D Help Manual. (Note that MAC computers already have a built-in 2D/3D grapher.)

**Supplemental Resources (optional)**

Paul's OL Notes - Calc III: The 3D Coordinate System

Patrick JMT Just Math video tutorials: PatrickJMT.com/#calculus

I'm not recommending this site today since the vector examples tend to be 2D, not 3D. Instead see ...

James Sousa's Math Is Power 4U video tutorials: MathIsPower4U - Calc II

Scroll down the third column to the "Vectors in Space" category. The first 10+ vectors listed are useful for this unit. Stop at "Dot Product" which is covered in our next unit.