Analytical Geometry \& Linear Algebra -- I

Course name: Analytical Geometry \& Linear Algebra -- I
Course number: XYZ

Course Characteristics

Key concepts of the class

fundamental principles of vector algebra,
concepts of basic geometry objects and their transformations in the plane and in the space

What is the purpose of this course?

This is an introductory course in analytical geometry and linear algebra. After having studied the course, students get to know fundamental principles of vector algebra and its applications in solving various geometry problems, different types of equations of lines and planes, conics and quadric surfaces, transformations in the plane and in the space. An introduction on matrices and determinants as a fundamental knowledge of linear algebra is also provided.

Course objectives based on Bloom’s taxonomy

- What should a student remember at the end of the course?

By the end of the course, the students should be able to

List basic notions of vector algebra,
recite the base form of the equations of transformations in planes and spaces,
recall equations of lines and planes,
identify the type of conic section,
recognize the kind of quadric surfaces.

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to

explain the geometrical interpretation of the basic operations of vector algebra,
restate equations of lines and planes in different forms,
interpret the geometrical meaning of the conic sections in the mathematical expression,
give the examples of the surfaces of revolution,
understand the value of geometry in various fields of science and techniques.

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to

Perform the basic operations of vector algebra,
use different types of equations of lines and planes to solve the plane and space problems,
represent the conic section in canonical form,
compose the equation of quadric surface.

Course evaluation

Course grade breakdown
Type	Points
Labs/seminar classes	10
Interim performance assessment	20
Exams	70

Grades range

Course grading range
Grade	Points
A	[80, 100]
B	[60, 79]
C	[40, 59]
D	[0, 39]

Resources and reference material

\bibentry{Sharipov:2013}

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Section 1

Section title

Vector algebra

Topics covered in this section

Vector spaces
Basic operations on vectors (summation, multiplication by scalar, dot product)
Linear dependency and in-dependency of the vectors
Basis in vector spaces

What forms of evaluation were used to test students’ performance in this section?


Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	1
Discussions	1

Typical questions for ongoing performance evaluation within this section

How to perform the shift of the vector?
What is the geometrical interpretation of the dot product?
How to determine whether the vectors are linearly dependent?
What is a vector basis?

Typical questions for seminar classes (labs) within this section

Evaluate $|{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}$ given that $|{\textbf {a}}|=4$ , $|{\textbf {b}}|=1$ , $\angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }$ .
Prove that vectors ${\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})$ and ${\textbf {a}}$ are perpendicular to each other.
Bases $AD$ and $BC$ of trapezoid $ABCD$ are in the ratio of $4:1$ . The diagonals of the trapezoid intersect at point $M$ and the extensions of sides $AB$ and $CD$ intersect at point $P$ . Let us consider the basis with $A$ as the origin, ${\overrightarrow {AD}}$ and ${\overrightarrow {AB}}$ as basis vectors. Find the coordinates of points $M$ and $P$ in this basis.
A line segment joining a vertex of a tetrahedron with the centroid of the opposite face (the centroid of a triangle is an intersection point of all its medians) is called a median of this tetrahedron. Using vector algebra prove that all the four medians of any tetrahedron concur in a point that divides these medians in the ratio of $3:1$ , the longer segments being on the side of the vertex of the tetrahedron.