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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

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

Typical questions for seminar classes (labs) within this section

1. Evaluate $|{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}$ given that $|{\textbf {a}}|=4$, $|{\textbf {b}}|=1$, $\angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }$.
2. 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.
3. 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.
4. 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.

Test questions for final assessment in this section

1. Vector spaces. General concepts.
2. Dot product as an operation on vectors.
3. Basis in vector spaces. Its properties.

Section 2

Section title

Introduction to matrices and determinants

Topics covered in this section

• Relationship between Linear Algebra and Analytical Geometry
• Matrices 2x2, 3x3
• Determinants 2x2, 3x3
• Operations om matrices and determinants
• The rank of a matrix
• Inverse matrix
• Systems of linear equations
• Changing basis and coordinates

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

Form	Yes/No
Development of individual parts of software product code	1
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

1. What is the difference between matrices and determinants?
2. Matrices $A$ and $C$ have dimensions of $m\times n$ and $p\times q$ respectively, and it is known that the product $ABC$ exists. What are possible dimensions of $B$ and $ABC$?
3. How to determine the rank of a matrix?
4. What is the meaning of the inverse matrix?
5. How to restate a system of linear equations in the matrix form?

Typical questions for seminar classes (labs) within this section

1. Find $A+B$ and $2A-3B+I$.
2. Find the products $AB$ and $BA$ (and so make sure that, in general, $AB\neq BA$ for matrices).
3. Find the inverse matrices for the given ones.
4. Find the determinants of the given matrices.
5. Point $M$ is the centroid of face $BCD$ of tetrahedron $ABCD$. The old coordinate system is given by $A$, ${\overrightarrow {AB}}$, ${\overrightarrow {AC}}$, ${\overrightarrow {AD}}$, and the new coordinate system is given by $M$, ${\overrightarrow {MB}}$, ${\overrightarrow {MC}}$, ${\overrightarrow {MA}}$. Find the coordinates of a point in the old coordinate system given its coordinates $x'$, $y'$, $z'$ in the new one.

Test questions for final assessment in this section

1. Operations om matrices and determinants.
2. Inverse matrix.
3. Systems of linear equations and their solution in matrix form.
4. Changing basis and coordinates.

Section 3

Section title

Lines in the plane and in the space

Topics covered in this section

• General equation of a line in the plane
• General parametric equation of a line in the space
• Line as intersection between planes
• Vector equation of a line
• Distance from a point to a line
• Distance between lines
• Inter-positioning of lines

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

1. How to represent a line in the vector form?
2. What is the result of intersection of two planes in vector form?
3. How to derive the formula for the distance from a point to a line?
4. How to interpret geometrically the distance between lines?
5. List all possible inter-positions of lines in the space.

Typical questions for seminar classes (labs) within this section

1. Two lines are given by the equations ${\textbf {r}}\cdot {\textbf {n}}=A$ and ${\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t$, and at that ${\textbf {a}}\cdot {\textbf {n}}\neq 0$. Find the position vector of the intersection point of these lines.
2. Find the distance from point $M_{0}$ with the position vector ${\textbf {r}}_{0}$ to the line defined by the equation (a) ${\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t$; (b) ${\textbf {r}}\cdot {\textbf {n}}=A$.
3. Diagonals of a rhombus intersect at point $M(1;\,2)$, the longest of them being parallel to a horizontal axis. The side of the rhombus equals 2 and its obtuse angle is $120^{\circ }$. Compose the equations of the sides of this rhombus.
4. Compose the equations of lines passing through point $A(2;-4)$ and forming angles of $60^{\circ }$ with the line ${\frac {1-2x}{3}}={\frac {3+2y}{-2}}$.

Test questions for final assessment in this section

1. Lines in the plane and in the space. Equations of lines.
2. Distance from a point to a line.
3. Distance between two parallel lines.
4. Distance between two skew lines.