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| Question || An ellipsoid is symmetric with respect to coordinate planes, passes through point <math>{\textstyle M(3;\,1;\,1)}</math> and circle <math>{\textstyle x^{2}+y^{2}+z^{2}=9}</math> , <math>{\textstyle x-z=0}</math> Find the equation of this ellipsoid || 0
 
| Question || An ellipsoid is symmetric with respect to coordinate planes, passes through point <math>{\textstyle M(3;\,1;\,1)}</math> and circle <math>{\textstyle x^{2}+y^{2}+z^{2}=9}</math> , <math>{\textstyle x-z=0}</math> Find the equation of this ellipsoid || 0
 
|}
 
|}
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=== Final assessment ===
  +
'''Section 1'''
  +
# Vector spaces General concepts
  +
# Dot product as an operation on vectors
  +
# Basis in vector spaces Its properties
  +
'''Section 2'''
  +
# Operations om matrices and determinants
  +
# Inverse matrix
  +
# Systems of linear equations and their solution in matrix form
  +
# Changing basis and coordinates
  +
'''Section 3'''
  +
# Lines in the plane and in the space Equations of lines
  +
# Distance from a point to a line
  +
# Distance between two parallel lines
  +
# Distance between two skew lines
  +
'''Section 4'''
  +
# Planes in the space Equations of planes
  +
# Distance from a point to a plane, from a line to a plane
  +
# Projection of a vector on the plane
  +
# Cross product, its properties and geometrical interpretation
  +
# Scalar triple product, its properties and geometrical interpretation
  +
'''Section 5'''
  +
# Determine the type of a given curve with the use of the method of invariant
  +
# Compose the canonical equation of a given curve
  +
# Determine the canonical coordinate system for a given curve
  +
'''Section 6'''
  +
# Determine the type of a quadric surface given by a certain equation
  +
# Compose the equation of a surface of revolution with the given directrix and generatrix
  +
# Represent a given equation of a quadric surface in the vector form
  +
  +
=== The retake exam ===
  +
'''Section 1'''
  +
  +
'''Section 2'''
  +
  +
'''Section 3'''
  +
  +
'''Section 4'''
  +
  +
'''Section 5'''
  +
  +
'''Section 6'''

Revision as of 18:36, 19 April 2022

Analytical Geometry & Linear Algebra – I

  • Course name: Analytical Geometry & Linear Algebra – I
  • Code discipline:
  • Subject area: ['fundamental principles of vector algebra,', 'concepts of basic geometry objects and their transformations in the plane and in the space']

Short Description

Prerequisites

Prerequisite subjects

Prerequisite topics

Course Topics

Course Sections and Topics
Section Topics within the section
Vector algebra
  1. Vector spaces
  2. Basic operations on vectors (summation, multiplication by scalar, dot product)
  3. Linear dependency and in-dependency of the vectors
  4. Basis in vector spaces
Introduction to matrices and determinants
  1. Relationship between Linear Algebra and Analytical Geometry
  2. Matrices 2x2, 3x3
  3. Determinants 2x2, 3x3
  4. Operations om matrices and determinants
  5. The rank of a matrix
  6. Inverse matrix
  7. Systems of linear equations
  8. Changing basis and coordinates
Lines in the plane and in the space
  1. General equation of a line in the plane
  2. General parametric equation of a line in the space
  3. Line as intersection between planes
  4. Vector equation of a line
  5. Distance from a point to a line
  6. Distance between lines
  7. Inter-positioning of lines
Planes in the space
  1. General equation of a plane
  2. Normalized linear equation of a plane
  3. Vector equation of a plane
  4. Parametric equation a plane
  5. Distance from a point to a plane
  6. Projection of a vector on the plane
  7. Inter-positioning of lines and planes
  8. Cross Product of two vectors
  9. Triple Scalar Product
Quadratic curves
  1. Circle
  2. Ellipse
  3. Hyperbola
  4. Parabola
  5. Canonical equations
  6. Shifting of coordinate system
  7. Rotating of coordinate system
  8. Parametrization
Quadric surfaces
  1. General equation of the quadric surfaces
  2. Canonical equation of a sphere and ellipsoid
  3. Canonical equation of a hyperboloid and paraboloid
  4. Surfaces of revolution
  5. Canonical equation of a cone and cylinder
  6. Vector equations of some quadric surfaces

Intended Learning Outcomes (ILOs)

What is the main 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.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

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.

Level 2: What basic practical skills should a student be able to perform?

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.

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

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.

Grading

Course grading range

Grade Range Description of performance
A. Excellent 80-100 -
B. Good 60-79 -
C. Satisfactory 40-59 -
D. Poor 0-39 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Labs/seminar classes 10
Interim performance assessment 20
Exams 70

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

Closed access resources

Software and tools used within the course

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section
Learning Activities Section 1 Section 2 Section 3 Section 4 Section 5 Section 6
Homework and group projects 1 1 1 1 1 1
Midterm evaluation 1 1 1 1 1 1
Testing (written or computer based) 1 1 1 1 1 1
Discussions 1 1 1 1 1 1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Question How to perform the shift of the vector? 1
Question What is the geometrical interpretation of the dot product? 1
Question How to determine whether the vectors are linearly dependent? 1
Question What is a vector basis? 1
Question Evaluate given that , , 0
Question Prove that vectors and are perpendicular to each other 0
Question Bases and of trapezoid are in the ratio of The diagonals of the trapezoid intersect at point and the extensions of sides and intersect at point Let us consider the basis with as the origin, and as basis vectors Find the coordinates of points and in this basis 0
Question 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 , the longer segments being on the side of the vertex of the tetrahedron 0

Section 2

Activity Type Content Is Graded?
Question What is the difference between matrices and determinants? 1
Question Matrices and have dimensions of and respectively, and it is known that the product exists What are possible dimensions of and  ? 1
Question How to determine the rank of a matrix? 1
Question What is the meaning of the inverse matrix? 1
Question How to restate a system of linear equations in the matrix form? 1
Question Find and 0
Question Find the products and (and so make sure that, in general, for matrices) 0
Question Find the inverse matrices for the given ones 0
Question Find the determinants of the given matrices 0
Question Point is the centroid of face of tetrahedron The old coordinate system is given by , , , , and the new coordinate system is given by , , , Find the coordinates of a point in the old coordinate system given its coordinates , , in the new one 0

Section 3

Activity Type Content Is Graded?
Question How to represent a line in the vector form? 1
Question What is the result of intersection of two planes in vector form? 1
Question How to derive the formula for the distance from a point to a line? 1
Question How to interpret geometrically the distance between lines? 1
Question List all possible inter-positions of lines in the space 1
Question Two lines are given by the equations and , and at that Find the position vector of the intersection point of these lines 0
Question Find the distance from point with the position vector to the line defined by the equation (a)  ; (b) 0
Question Diagonals of a rhombus intersect at point , the longest of them being parallel to a horizontal axis The side of the rhombus equals 2 and its obtuse angle is Compose the equations of the sides of this rhombus 0
Question Compose the equations of lines passing through point and forming angles of with the line 0

Section 4

Activity Type Content Is Graded?
Question What is the difference between general and normalized forms of equations of a plane? 1
Question How to rewrite the equation of a plane in a vector form? 1
Question What is the normal to a plane? 1
Question How to interpret the cross products of two vectors? 1
Question What is the meaning of scalar triple product of three vectors? 1
Question Find the cross product of (a) vectors and  ; (b) vectors and 0
Question A triangle is constructed on vectors and (a) Find the area of this triangle (b) Find the altitudes of this triangle 0
Question Find the scalar triple product of , , 0
Question It is known that basis vectors , , have lengths of , , respectively, and , , Find the volume of a parallelepiped constructed on vectors with coordinates , and in this basis 0

Section 5

Activity Type Content Is Graded?
Question Formulate the canonical equation of the given quadratic curve 1
Question Which orthogonal transformations of coordinates do you know? 1
Question How to perform a transformation of the coordinate system? 1
Question How to represent a curve in the space? 1
Question Prove that a curve given by 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 0
Question Determine types of curves given by the following equations For each of the curves, find its canonical coordinate system (ie indicate the coordinates of origin and new basis vectors in the initial coordinate system) and its canonical equation (a)  ; (b)  ; (c)  ; (d)  ; 0
Question Find the equations of lines tangent to curve that are (a) parallel to line  ; (b) perpendicular to line  ; (c) parallel to line 0

Section 6

Activity Type Content Is Graded?
Question What is the type of a quadric surface given by a certain equation? 1
Question How to compose the equation of a surface of revolution? 1
Question What is the difference between a directrix and generatrix? 1
Question How to represent a quadric surface in the vector form? 1
Question For each value of parameter determine types of surfaces given by the equations: (a)  ; (b)  ; (c)  ; (d) 0
Question Find a vector equation of a right circular cone with apex and axis if it is known that generatrices of this cone form the angle of with its axis 0
Question Find the equation of a cylinder with radius that has an axis , , 0
Question An ellipsoid is symmetric with respect to coordinate planes, passes through point and circle , Find the equation of this ellipsoid 0

Final assessment

Section 1

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

Section 2

  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

  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

  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

  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

  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

The retake exam

Section 1

Section 2

Section 3

Section 4

Section 5

Section 6