Premium Only Content

Vectors and the Geometry of Space: Equations of Lines and Planes
In this video I go over equations of lines and planes in 3D coordinates. This section is part of my calculus book chapter titled Vectors and the Geometry of Space. In 2D coordinates, a line can be described by a point on the line and its slope or direction. Similarly, a line in 3D coordinates can be described by a point on the line and a vector parallel to it which thus serves as the direction. A plane in 3D coordinates is described by a point on the plane and a vector perpendicular to the plane. The plane equation utilizes the fact that the dot product of two perpendicular vectors is equal to zero. Furthermore, the cross product is also used in determining the perpendicular vector, termed a normal vector. I go through several related concepts, such as scalar and vector equations of lines, segments of lines, and linear equations of planes, as well as many examples to give a thorough examination of 3D lines and planes.
The topics covered as well as their timestamps are listed below.
- Introduction: 0:00
- Calculus Book Reference: 1:13
- Sections in Calculus Book Chapter: 1:27
- Topics to Cover: 2:28
1. Lines: 3:42
- Vector Equation of a Line: 5:00
- Parametric Equations of a Line: 12:50
- Example 1: 17:22
- Solution to 1(a): 17:49
- Solution to 1(b): 25:55
- Direction Numbers: 28:08
- Symmetric Equations of a Line: 31:49
- Example 2: 37:29
- Solution to 2(a): 37:54
- Solution to 2(b): 43:43
- Note on Example 2: 48:38
- Equation of a Line Segment: 50:24
- Example 3: 1:04:16
2. Planes: 1:15:15
- Vector Equation of a Plane: 1:15:32
- Scalar Equation of a Plane: 1:21:13
- Example 4: 1:23:14
- Linear Equation of a Plane: 1:29:40
- Example 5: 1:32:58
- Example 6: 1:42:27
- Parallel Planes: 1:48:18
- Example 7: 1:52:34
- Solution to 7(a): 1:52:58
- Solution to 7(b): 1:58:13
- Note 1: Two Linear Equations Can Represent a Line: 2:08:20
- Note 2: Line of Intersection from Two Parametric Planes: 2:16:18
- Example 8: Distance from a Point to a Plane: 2:19:55
- Example 9: 2:30:45
- Example 10: 2:38:04
- Exercise 1: 2:55:19
- PDF video notes: https://1drv.ms/b/s!As32ynv0LoaIiecyXiOIVbA_nizjLQ
- HIVE video notes: https://peakd.com/hive-128780/@mes/equations-of-lines-and-planes
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FO3u0IupqllNNkZTsffpIV
- Vectors and the Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ
Related Videos:
Discovery Project: The Geometry of a Tetrahedron: https://peakd.com/hive-128780/@mes/discovery-project-geometry-of-a-tetrahedron
The Cross Product: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product
The Dot Product: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-dot-product
Complex Numbers as Rotation Matrices: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices
Vectors: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors
3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems .
------------------------------------------------------
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
Like, Subscribe, Favorite, and Comment Below!
Follow us on:
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: https://peakd.com/@mes
MORE Links: https://linktr.ee/matheasy
Email me: contact@mes.fm
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
-
2:15:54
Math Easy Solutions
8 days ago $0.04 earnedMES Math Q/A 31: What is a Potentiometer?
2271 -
UPCOMING
Ben Shapiro
1 hour agoEp. 2264 - Democratic Party BLEEDING Voters!
1.85K2 -
LIVE
Timcast
1 hour agoTrump Civil Fraud Judgement OVERTURNED On Appeal, PROVING Corruption
24,911 watching -
LIVE
The Charlie Kirk Show
1 hour agoTrump the Unbreakable + The Anti-Abstraction Presidency | Marlow, Morris, Hanson, Nuclear | 8.21.25
3,249 watching -
Steven Crowder
3 hours ago🔴 Woke CNN Host's Meltdown Over Trump Slavery Truth Needs to Be Examined
188K154 -
19:42
Law&Crime
21 hours agoPraying Florida Teen Pleads Not Guilty After Parents’ Gruesome Murder
95 -
LIVE
JuicyJohns
4 hours ago $1.23 earned🟢#1 REBIRTH PLAYER 10.2+ KD🟢
296 watching -
1:00:50
VINCE
4 hours agoThe Pro-Crime Left Is Officially Finding Out | Episode 108 - 08/21/25
155K171 -
44:15
Nikko Ortiz
2 hours agoLive - News, Politics, Podcast And Naaah Im Playin We Chillen
17.7K2 -
1:56:55
Dear America
4 hours agoLUNATICS Protest DC Crime Being Down 30%?! + FBI Catches A TOP MOST WANTED!!
97.4K47