Chapter 9: Relationships between Points, Lines, and Planes
Strand: Geometry and Algebra of Vectors
V3. distinguish between the geometric representations of a single linear equation or a system of two linear equations in
two-space and three-space, and determine different geometric configurations of lines and planes in three-space
V4. represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and
intersections.
Optional Preparation/Review for Lines and Planes
• Do: pg. 487 #1b,d, 2b,f, 3a, 4, 5, 6b, 7, 8
Note: Page 486 summarizes the knowledge base required for chapter 9. Since it is essentially a
summary of chapter 8, we will not spend class time doing this work.
9.1 Intersection of Two Lines in ℝ2 and ℝ3 and Intersection of a Line and a Plane in ℝ3
• Learn terminology and methods for the intersection of two line and a line with a plane.
• Learn terms: skew lines, inconsistent system of linear equations (a.k.a. Inconsistent system or
inconsistent SOLE), consistent system, independent solution, and dependent solution.
• Do: Chapter 9.1 pg. 496, 497 #1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15
Notes:
#1. Know the three ways two lines relate in ℝ2 and the ways to determine the relationship. Know the
importance of slopes (or direction vectors). Using a grade 10 method such as elimination and
substitution on the Cartesian equations is usually the easiest. Equating paramatrics is an alternative.
#2. Know the four ways two lines can relate in ℝ3 . Master the method of equating parametric
equations, solving for the parameters, and then substituting the parameters to determine any
intersection. Know the importance of direction vectors.
#3. Know the three ways a line and a plane can relate in ℝ3 and the ways to determine these
relationships. Know both methods for determining an intersection: equating parametric equations (see
#2) and substituting the paramatric equation of a line into the Cartesian equation of the plane.
#4. Know the terms: skew lines, inconsistent system of linear equations (a.k.a. inconsistent system or
inconsistent SOLE), consistent system, independent solution, and dependent solution – can also be
done in 9.2 (pg. 504)
9.2 Solving Systems of Equations
• Extend the grade 10 method of elimination in order to solve larger systems of linear equations.
• Do Chapter 9.2 pg. 507 to 509 #1, 2, 3, 7, 8, 9, 10, 12, 13, 14
Notes:
#1. Know the three elementary row operations (EROs): multiplying an equation by a non-zero
constant, interchanging (or switching) any two equation in the system, and adding (or subracting) one
equation to a second one (and replacing the second one with the resulting equation). Note:
Combining EROs is permitted. For example, multiplying an equation and subracting it from another.
#2. Recognize when a SOLE has a solution and when it does not (consistent or inconsistent).
#3. Determine solutions to a SOLE using EROs in an organized fashion.
#4. PRACTICE, PRACTICE, PRACTICE
9.3 The Intersection of Two Planes in ℝ3
• Use skills for solving systems of equations (9.2) in order to determine the relationship between
two planes: parallel planes, coincident planes or planes that intersect in a line.
• Do Chapter 9.3 pg. 516, 517 #1 to 11
Notes:
#1. A suggestion is to have this lesson consist of working through the assigned work as a class.
#2. Make notes in your solutions will likely be helpful as connecting the equations to the geometric
interpretation is crucial.
#3. Examining normals of planes is important in determining the relationship between the given
planes. Therefore, represent planes by Cartesian equations. Unlike the questions in this section,
which provide Cartesian equations, future questions may require you to convert to Cartesian form.
Mid-Chapter Review:
This work is not assigned at this time. Please note that doing these later in the course is an excellent
way to study for the final exam.
Matrices: Vector Appendix
• Use the basics learned in 9.2 Solving Systems of Equations in a new form: Matrices.
• Use Gaussian Elimination (Example 3, pg. 587), Gauss-Jordan Elimination (Example 4, pg.
593, 594) and the graphing calculator to solve SOLEs.
• Do Exercise pg. 588, 589 #1, 4, 6, 8, 10 (do not interpret #10 geometrically yet)
• Do Exercise pg. 595 #3, 4, 5
Notes:
#1. Knowledge of matrix terms is essential: elements, diagonals, principal or main diagonal, rows,
columns, dimensions, coefficient matrix (pg. 583), augmented matrix (pg. 583), answer matrix or
answer vector, elementary row operations (pg. 584), Row-Echelon Form (REF) (pg. 584, 586),
Reduced Row-Echelon Form (RREF) (pg. 590), and identity matrix (pg. 594).
#2. Introduction of matrix notation is desired but not required (i.e. A m x n=[a ij ] where A is the variable
for the entire matrix, m is the number of rows, n is the number of columns, a represents the elements
in the matrix, i is the row number for the element, and j is the column number for the element).
#3. We will go over how to solve a SOLE using MATRIX on the graphing calculator.
#4. Note examples listed above: Example 3, pg. 587 and Example pg. 593, 594.
9.4 The Relationships of Three Planes to Each Other
• Use an analysis of normals and methods for solving systems of equations (especially matrices)
to determine the relationship among three planes and all points of intersections.
• Do pg. 530 to 533 #1 to 6 with a focus on understanding the connection between the equations
and the geometric interpretation, #8 to 13 with the graphing calculator. Give an geometric
interpretation to the solutions of pg. 584 #10 (done previously).
Notes:
#1. Know the eight ways three planes can be related to each other: 3 coincident planes, 2 coincident
planes with a parallel plane, 3 parallel planes, 2 coincident planes with a non-parallel plane, 2 parallel
planes with a non-parallel plane, 3 planes that intersect in an infinite triangular prism, 3 planes that
intersect in a line, and 3 planes that intersect in a point.
#2. Work at being proficient at determining which of the eight situations is true in a given problem or a
given SOLE.
9.5 Distance from a Point to Line in ℝ2 and ℝ3
9.6 Distance from a Point to Plane in ℝ3
• Understand the development of the three distance formulas.
• Use the distance formulas.
• Do pg. 540, 541 #5, 6, 7, 8
• Do pg. 550 #2, 3, 5
Notes:
#1. The IN SUMMARY on pg. 539 lists the formulas that students need to memorize.
#2. The IN SUMMARY on pg. 549 lists the formula students need to memorize and two methods used
to determine the distance between skew lines.
The Practice Test found on page 556 is excellent preparation for the vectors exam. Of course, you
should do #4 and 5 by matrices to get full preparation value.
Summative Task
Presentation of Learning (selected students; everyone will need to do at least one in this course)
Instead of a written Unit Test, the Chapter 9 material will incorporated into the Vectors Exam.
(Communication may be graded using the rubric adapted from the Achievement Chart: Mathematics, Grades 9–12.)
Communication. The conveying of meaning through various oral, written, and visual
forms (e.g., providing explanations of reasoning or justification of results orally or in
writing; communicating mathematical ideas and solutions in writing, using numbers
and algebraic symbols, and visually, using pictures, diagrams, charts, tables, graphs,
and concrete materials).
Rubric.
Type of Communication
Level of effectiveness shown by the student:
Expression and organization of
ideas and mathematical thinking
(e.g. clarity of expression, logical
organization, level of justification or
reasoning)
None or Little
0 to 49%
Limited
50 to 59%
Some
60 to 69%
Considerable
(average)
70 to 79%
High Degree
80 to 94%
Superior
95 to 100%
Use of conventions, vocabulary,
and terminology of the discipline
(e.g. terms, symbols, diagrams,
charts, graphs)
None or Little
0 to 49%
Limited
50 to 59%
Some
60 to 69%
Considerable
(average)
70 to 79%
High Degree
80 to 94%
Superior
95 to 100%