BSc: Analytic Geometry And Linear Algebra I

Analytical Geometry & Linear Algebra – I

• Course name: Analytical Geometry & Linear Algebra – I
• Course number: XYZ
• Subject area: Math

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.

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

• 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?

• 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?

• 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

		Proposed points
Labs/seminar classes	20	10
Interim performance assessment	30	20
Exams	50	70

Grades range

Course grading range

		Proposed range
A. Excellent	90-100	80-100
B. Good	75-89	60-79
C. Satisfactory	60-74	40-59
D. Poor	0-59	0-39

Course Sections

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

Course Sections

Section	Section Title	Lectures	Seminars	Self-study	Knowledge
Number		(hours)	(labs)		evaluation
1	Vector algebra	8	4	10	2
2	Introduction to matrices and determinants	8	4	10	1
3	Lines in the plane and in the space	8	4	10	2
4	Planes in the space	8	4	10	1
5	Quadratic curves	8	4	10	2
6	Quadric surfaces	8	4	10	2
Final examination					2

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?

	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	0
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 ${\textstyle |{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}}$ given that ${\textstyle |{\textbf {a}}|=4}$, ${\textstyle |{\textbf {b}}|=1}$, ${\textstyle \angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }}$.
2. Prove that vectors ${\textstyle {\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}$ and ${\textstyle {\textbf {a}}}$ are perpendicular to each other.
3. Bases ${\textstyle AD}$ and ${\textstyle BC}$ of trapezoid ${\textstyle ABCD}$ are in the ratio of ${\textstyle 4:1}$. The diagonals of the trapezoid intersect at point ${\textstyle M}$ and the extensions of sides ${\textstyle AB}$ and ${\textstyle CD}$ intersect at point ${\textstyle P}$. Let us consider the basis with ${\textstyle A}$ as the origin, ${\textstyle {\overrightarrow {AD}}}$ and ${\textstyle {\overrightarrow {AB}}}$ as basis vectors. Find the coordinates of points ${\textstyle M}$ and ${\textstyle P}$ 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 ${\textstyle 3:1}$, 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?

	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	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. What is the difference between matrices and determinants?
2. Matrices ${\textstyle A}$ and ${\textstyle C}$ have dimensions of ${\textstyle m\times n}$ and ${\textstyle p\times q}$ respectively, and it is known that the product ${\textstyle ABC}$ exists. What are possible dimensions of ${\textstyle B}$ and ${\textstyle ABC}$?
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

Let ${\textstyle A=\left({\begin{array}{cc}3&1\\5&-2\\\end{array}}\right)}$, ${\textstyle B=\left({\begin{array}{cc}-2&1\\3&4\\\end{array}}\right)}$, and ${\textstyle I=\left({\begin{array}{cc}1&0\\0&1\\\end{array}}\right)}$.

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

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

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.

Section 4

Section title:

Planes in the space

Topics covered in this section:

• General equation of a plane
• Normalized linear equation of a plane
• Vector equation of a plane
• Parametric equation a plane
• Distance from a point to a plane
• Projection of a vector on the plane
• Inter-positioning of lines and planes
• Cross Product of two vectors
• Triple Scalar Product

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

	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	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. What is the difference between general and normalized forms of equations of a plane?
2. How to rewrite the equation of a plane in a vector form?
3. What is the normal to a plane?
4. How to interpret the cross products of two vectors?
5. What is the meaning of scalar triple product of three vectors?

Typical questions for seminar classes (labs) within this section

1. Find the cross product of (a) vectors ${\textstyle {\textbf {a}}(3;-2;\,1)}$ and ${\textstyle {\textbf {b}}(2;-5;-3)}$; (b) vectors ${\textstyle {\textbf {a}}(3;-2;\,1)}$ and ${\textstyle {\textbf {c}}(-18;\,12;-6)}$.
2. A triangle is constructed on vectors ${\textstyle {\textbf {a}}(2;4;-1)}$ and ${\textstyle {\textbf {b}}(-2;1;1)}$. (a) Find the area of this triangle. (b) Find the altitudes of this triangle.
3. Find the scalar triple product of ${\textstyle {\textbf {a}}(1;\,2;-1)}$, ${\textstyle {\textbf {b}}(7;3;-5)}$, ${\textstyle {\textbf {c}}(3;\,4;-3)}$.
4. It is known that basis vectors ${\textstyle {\textbf {e}}_{1}}$, ${\textstyle {\textbf {e}}_{2}}$, ${\textstyle {\textbf {e}}_{3}}$ have lengths of ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 2{\sqrt {2}}}$ respectively, and ${\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{2})=120^{\circ }}$, ${\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{3})=135^{\circ }}$, ${\textstyle \angle ({\textbf {e}}_{2},{\textbf {e}}_{3})=45^{\circ }}$. Find the volume of a parallelepiped constructed on vectors with coordinates ${\textstyle (-1;\,0;\,2)}$, ${\textstyle (1;\,1\,4)}$ and ${\textstyle (-2;\,1;\,1)}$ in this basis.

Test questions for final assessment in this section

1. Planes in the space. Equations of planes.
2. Distance from a point to a plane, from a line to a plane.
3. Projection of a vector on the plane.
4. Cross product, its properties and geometrical interpretation.
5. Scalar triple product, its properties and geometrical interpretation.

Section 5

Section title:

Quadratic curves

Topics covered in this section:

• Circle
• Ellipse
• Hyperbola
• Parabola
• Canonical equations
• Shifting of coordinate system
• Rotating of coordinate system
• Parametrization

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

	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	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Formulate the canonical equation of the given quadratic curve.
2. Which orthogonal transformations of coordinates do you know?
3. How to perform a transformation of the coordinate system?
4. How to represent a curve in the space?

Typical questions for seminar classes (labs) within this section

1. Prove that a curve given by ${\textstyle 34x^{2}+24xy+41y^{2}-44x+58y+1=0}$ is an ellipse. Find the major and minor axes of this ellipse, its eccentricity, coordinates of its center and foci. Find the equations of axes and directrices of this ellipse.
2. Determine types of curves given by the following equations. For each of the curves, find its canonical coordinate system (i.e. indicate the coordinates of origin and new basis vectors in the initial coordinate system) and its canonical equation. (a) ${\textstyle 9x^{2}-16y^{2}-6x+8y-144=0}$; (b) ${\textstyle 9x^{2}+4y^{2}+6x-4y-2=0}$; (c) ${\textstyle 12x^{2}-12x-32y-29=0}$; (d) ${\textstyle xy+2x+y=0}$;
3. Find the equations of lines tangent to curve ${\textstyle 6xy+8y^{2}-12x-26y+11=0}$ that are (a) parallel to line ${\textstyle 6x+17y-4=0}$; (b) perpendicular to line ${\textstyle 41x-24y+3=0}$; (c) parallel to line ${\textstyle y=2}$.

Test questions for final assessment in this section

1. Determine the type of a given curve with the use of the method of invariant.
2. Compose the canonical equation of a given curve.
3. Determine the canonical coordinate system for a given curve.

Section 6

Section title:

Quadric surfaces

Topics covered in this section:

• General equation of the quadric surfaces
• Canonical equation of a sphere and ellipsoid
• Canonical equation of a hyperboloid and paraboloid
• Surfaces of revolution
• Canonical equation of a cone and cylinder
• Vector equations of some quadric surfaces

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

	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	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. What is the type of a quadric surface given by a certain equation?
2. How to compose the equation of a surface of revolution?
3. What is the difference between a directrix and generatrix?
4. How to represent a quadric surface in the vector form?

Typical questions for seminar classes (labs) within this section

1. For each value of parameter ${\textstyle a}$ determine types of surfaces given by the equations: (a) ${\textstyle x^{2}+y^{2}-z^{2}=a}$; (b) ${\textstyle x^{2}+a\left(y^{2}+z^{2}\right)=1}$; (c) ${\textstyle x^{2}+ay^{2}=az}$; (d) ${\textstyle x^{2}+ay^{2}=az+1}$.
2. Find a vector equation of a right circular cone with apex ${\textstyle M_{0}\left({\textbf {r}}_{0}\right)}$ and axis ${\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}$ if it is known that generatrices of this cone form the angle of ${\textstyle \alpha }$ with its axis.
3. Find the equation of a cylinder with radius ${\textstyle {\sqrt {2}}}$ that has an axis ${\textstyle x=1+t}$, ${\textstyle y=2+t}$, ${\textstyle z=3+t}$.
4. An ellipsoid is symmetric with respect to coordinate planes, passes through point ${\textstyle M(3;\,1;\,1)}$ and circle ${\textstyle x^{2}+y^{2}+z^{2}=9}$, ${\textstyle x-z=0}$. Find the equation of this ellipsoid.

Test questions for final assessment in this section

1. Determine the type of a quadric surface given by a certain equation.
2. Compose the equation of a surface of revolution with the given directrix and generatrix.
3. Represent a given equation of a quadric surface in the vector form.