Lecture Notes: Vector Spaces and 3D Coordinate Systems

Introduction

• Topic: Vector space in geometry, specifically focusing on 3D coordinate systems.
• Reference: Calculus book "Calculus: Early Transcendentals, 7th Edition" by James Stewart.

Topics Covered

1. Vectors and the geometry of space.
2. Three-dimensional coordinate systems.
3. Distance formula in three dimensions.
4. Equations of spheres.
5. Examples with solutions.

Vectors and Geometry of Space

• Introduction to vectors in 3D space.
• Vectors provide simple descriptions of lines, planes, surfaces, and solids.

Three-Dimensional Coordinate Systems

• Points in space are represented by ordered triples ((a, b, c)).
• Coordinate axes: x-axis, y-axis, z-axis, with the right-hand rule for orientation.
• Coordinate planes: xy-plane, yz-plane, xz-plane.
• Space is divided into eight parts called 'octants'.
• Visualizing 3D coordinates using room analogy.
• Points in space: (A) is the distance from the yz-plane, (B) from the xz-plane, and (C) from the xy-plane.

Distance Formula in 3D

• Formula: (\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2})
• Visual representation using rectangular boxes.

Equations of Spheres

• General equation: ((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2)
• Centered at ((h, k, l)) with radius (r).

Examples

Example 1

• Surfaces represented by equations like (z = 3) and (y = 5).

Example 2

• Surfaces in (R^3) represented by (x^2 + y^2 = 1).
• Illustration of horizontal cylinders.

Example 3

• Surface represented by (y = x).
• Represents a vertical plane in (R^3).

Example 4

• Distance between points ((2, -1, 7)) and ((1, -3, 5)).
• Calculation using the distance formula.

Example 5

• Equation of a sphere with center ((h, k, l)) and radius (r).
• Derivation using distance formula.

Example 6

• Identifying the center and radius of a sphere given an equation.
• Completing the square to rewrite in standard form.

Example 7

• Region in (R^3) represented by inequalities (1 \leq x^2 + y^2 + z^2 \leq 4) and (z \leq 0).
• Visualization of solid region between two spheres.

Conclusion

• Overview of the series on vectors and 3D geometry, with more to come in future videos.