BSc: Analytic Geometry And Linear Algebra I.f22
===============================================






Contents
--------


* [1 Analytical Geometry & Linear Algebra – I](#Analytical_Geometry_.26_Linear_Algebra_.E2.80.93_I)
	+ [1.1 Short Description](#Short_Description)
	+ [1.2 Course Topics](#Course_Topics)
	+ [1.3 Intended Learning Outcomes (ILOs)](#Intended_Learning_Outcomes_.28ILOs.29)
		- [1.3.1 ILOs defined at three levels](#ILOs_defined_at_three_levels)
			* [1.3.1.1 Level 1: What concepts should a student know/remember/explain?](#Level_1:_What_concepts_should_a_student_know.2Fremember.2Fexplain.3F)
			* [1.3.1.2 Level 2: What basic practical skills should a student be able to perform?](#Level_2:_What_basic_practical_skills_should_a_student_be_able_to_perform.3F)
			* [1.3.1.3 Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?](#Level_3:_What_complex_comprehensive_skills_should_a_student_be_able_to_apply_in_real-life_scenarios.3F)
	+ [1.4 Grading](#Grading)
		- [1.4.1 Course grading range](#Course_grading_range)
		- [1.4.2 Course activities and grading breakdown](#Course_activities_and_grading_breakdown)
		- [1.4.3 Recommendations for students on how to succeed in the course](#Recommendations_for_students_on_how_to_succeed_in_the_course)
	+ [1.5 Resources, literature and reference materials](#Resources.2C_literature_and_reference_materials)
		- [1.5.1 Open access resources](#Open_access_resources)
	+ [1.6 Activities and Teaching Methods](#Activities_and_Teaching_Methods)
	+ [1.7 Formative Assessment and Course Activities](#Formative_Assessment_and_Course_Activities)
		- [1.7.1 Ongoing performance assessment](#Ongoing_performance_assessment)
			* [1.7.1.1 Section 1](#Section_1)
			* [1.7.1.2 Section 2](#Section_2)
			* [1.7.1.3 Section 3](#Section_3)
		- [1.7.2 Final assessment](#Final_assessment)
			* [1.7.2.1 Section 1](#Section_1_2)
			* [1.7.2.2 Section 2](#Section_2_2)
			* [1.7.2.3 Section 3](#Section_3_2)
		- [1.7.3 The retake exam](#The_retake_exam)



Analytical Geometry & Linear Algebra – I
========================================


* **Course name**: Analytical Geometry & Linear Algebra – I
* **Code discipline**: CSE202
* **Subject area**: Math


Short Description
-----------------


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 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 independency of the vectors. Basis in vector spaces.
4. Introduction to matrices and determinants. The rank of a matrix. Inverse matrix.
5. Systems of linear equations
6. Changing basis and coordinates
 |
| Line and Plane | 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. Distance between lines
6. General equation of a plane.
7. Normalized linear equation of a plane.
8. Vector equation of a plane. Parametric equation of a plane
9. Inter-positioning of lines and planes
10. Cross Product of two vectors. Triple Scalar Product
 |
| Quadratic curves and surfaces | 1. Circle, Ellipse, Hyperbola, Parabola. Canonical equations
2. Shift of coordinate system. Rotation of coordinate system. Parametrization
3. General equation of the quadric surfaces.
4. Canonical equations of a sphere, ellipsoid, hyperboloid and paraboloid
5. Surfaces of revolution. Canonical equation of a cone and cylinder
6. Vector equations of some quadric surfaces
 |


Intended Learning Outcomes (ILOs)
---------------------------------


### ILOs defined at three levels


We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.



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


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


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.


#### 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 ...



* 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.


Grading
-------


### Course grading range





| Grade | Range | Description of performance
 |
| --- | --- | --- |
| A. Excellent | 85-100 | -
 |
| B. Good | 70-84 | -
 |
| C. Satisfactory | 55-70 | -
 |
| D. Fail | 0-54 | -
 |


### Course activities and grading breakdown





| Activity Type | Percentage of the overall course grade
 |
| --- | --- |
| Midterm | 35
 |
| Tests | 30 (15 for each)
 |
| Final exam | 35
 |


### Recommendations for students on how to succeed in the course


* Participation is important. Attending lectures is the key to success in this course.
* Review lecture materials before classes to do well.
* Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.


Resources, literature and reference materials
---------------------------------------------


### Open access resources


* V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. Textbook. Tomsk: TPU Press, 2009, 114 pp [book1](https://portal.tpu.ru/SHARED/k/KONVAL/Textbooks/Tab1/Konev-Linear_Algebra_Vector_Algebra_and_Analytical_Geome.pdf)
* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [book2](https://arxiv.org/pdf/1111.6521.pdf)
* P.R. Vital. Analytical Geometry 2D and 3D Analytical Geometry 2D and 3D [book3](https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604)


Activities and Teaching Methods
-------------------------------




Teaching and Learning Methods within each section
| Teaching Techniques | Section 1 | Section 2 | Section 3
 |
| --- | --- | --- | --- |
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) | 1 | 1 | 1
 |
| Project-based learning (students work on a project) | 0 | 0 | 0
 |
| Modular learning (facilitated self-study) | 0 | 0 | 0
 |
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) | 1 | 1 | 1
 |
| Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) | 0 | 0 | 0
 |
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) | 0 | 0 | 0
 |
| Inquiry-based learning | 0 | 0 | 0
 |
| Just-in-time teaching | 0 | 0 | 0
 |
| Process oriented guided inquiry learning (POGIL) | 0 | 0 | 0
 |
| Studio-based learning | 0 | 0 | 0
 |
| Universal design for learning | 0 | 0 | 0
 |
| Task-based learning | 0 | 0 | 0
 |




Activities within each section
| Learning Activities | Section 1 | Section 2 | Section 3
 |
| --- | --- | --- | --- |
| Lectures | 1 | 1 | 1
 |
| Interactive Lectures | 1 | 1 | 1
 |
| Lab exercises | 1 | 1 | 1
 |
| Experiments | 0 | 0 | 0
 |
| Modeling | 0 | 0 | 0
 |
| Cases studies | 0 | 0 | 0
 |
| Development of individual parts of software product code | 0 | 0 | 0
 |
| Individual Projects | 0 | 0 | 0
 |
| Group projects | 0 | 0 | 0
 |
| Flipped classroom | 0 | 0 | 0
 |
| Quizzes (written or computer based) | 1 | 1 | 1
 |
| Peer Review | 0 | 0 | 0
 |
| Discussions | 1 | 1 | 1
 |
| Presentations by students | 0 | 0 | 0
 |
| Written reports | 0 | 0 | 0
 |
| Simulations and role-plays | 0 | 0 | 0
 |
| Essays | 0 | 0 | 0
 |
| Oral Reports | 0 | 0 | 0
 |


Formative Assessment and Course Activities
------------------------------------------


### Ongoing performance assessment


#### Section 1


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?
5. What is the difference between matrices and determinants?
6. Matrices 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31) and 



C


{\textstyle C}

![{\textstyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b) have dimensions of 



m
×
n


{\textstyle m\times n}

![{\textstyle m\times n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37dc29fae3b1932f1b311a052ecc6ecb8692dc48) and 



p
×
q


{\textstyle p\times q}

![{\textstyle p\times q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbd208094b241eb647c2e3b515a05197f9e0fdc) respectively, and it is known that the product 



A
B
C


{\textstyle ABC}

![{\textstyle ABC}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1516a55703463b2e378eb0b2eda76f08f6919636) exists. What are possible dimensions of 



B


{\textstyle B}

![{\textstyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0b47ffc21636dc2df68f6c793177a268f10e9b) and 



A
B
C


{\textstyle ABC}

![{\textstyle ABC}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1516a55703463b2e378eb0b2eda76f08f6919636)?
7. How to determine the rank of a matrix?
8. What is the meaning of the inverse matrix?
9. How to restate a system of linear equations in the matrix form?


#### Section 2


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.
6. What is the difference between general and normalized forms of equations of a plane?
7. How to rewrite the equation of a plane in a vector form?
8. What is the normal to a plane?
9. How to interpret the cross products of two vectors?
10. What is the meaning of scalar triple product of three vectors?


#### Section 3


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?
5. What is the type of a quadric surface given by a certain equation?
6. How to compose the equation of a surface of revolution?
7. What is the difference between a directrix and generatrix?
8. How to represent a quadric surface in the vector form?


### Final assessment


#### Section 1


1. Evaluate 




|



a




|


2


−
2


3




a


⋅


b


−
7

|



b




|


2




{\textstyle |{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}}

![{\textstyle |{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b39de16745ae08f1e071202ed4c68a7439df361b) given that 




|



a



|

=
4


{\textstyle |{\textbf {a}}|=4}

![{\textstyle |{\textbf {a}}|=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dad653ed7b4f53967e38f14dc254a0b4ed40f4ac), 




|



b



|

=
1


{\textstyle |{\textbf {b}}|=1}

![{\textstyle |{\textbf {b}}|=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2147d16bdc4fd215671610f46300d16ea8f1963d), 



∠
(


a


,



b


)
=

150

∘




{\textstyle \angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }}

![{\textstyle \angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34c0c47f4b2dbe58c362811be01e2a9658ffd65e).
2. Prove that vectors 





b


(


a


⋅


c


)
−


c


(


a


⋅


b


)


{\textstyle {\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}

![{\textstyle {\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91d497e3670b2e628a97f0ccb57ca80a925ef32c) and 





a




{\textstyle {\textbf {a}}}

![{\textstyle {\textbf {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7a4f8c41fd49715b57c7891e867192892aaf8b) are perpendicular to each other.
3. Bases 



A
D


{\textstyle AD}

![{\textstyle AD}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b67c9a22c95905ef6b81962b11b31e1c0ef52225) and 



B
C


{\textstyle BC}

![{\textstyle BC}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9357a907256f2d11e87040067a6ce9dd1976f725) of trapezoid 



A
B
C
D


{\textstyle ABCD}

![{\textstyle ABCD}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56e9dd14b4fcb989c106f3679aa2699f07eee6d4) are in the ratio of 



4
:
1


{\textstyle 4:1}

![{\textstyle 4:1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44785771746a94fd2cf3d26ff7cd5653f816763d). The diagonals of the trapezoid intersect at point 



M


{\textstyle M}

![{\textstyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088) and the extensions of sides 



A
B


{\textstyle AB}

![{\textstyle AB}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac121a3e6c5dff2ebb2ff8340e4a7f1b35992e44) and 



C
D


{\textstyle CD}

![{\textstyle CD}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523217c283f1b3f72fe4f317b8ff09e22365f1e5) intersect at point 



P


{\textstyle P}

![{\textstyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038590207af1024a629c1a08c855e9ac46bf5610). Let us consider the basis with 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31) as the origin, 






A
D

→




{\textstyle {\overrightarrow {AD}}}

![{\textstyle {\overrightarrow {AD}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a09091c3ae3ae05dbb915b79e411ee15a84401f1) and 






A
B

→




{\textstyle {\overrightarrow {AB}}}

![{\textstyle {\overrightarrow {AB}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9e6559e7cd7c16144340887587c73e294b4ba) as basis vectors. Find the coordinates of points 



M


{\textstyle M}

![{\textstyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088) and 



P


{\textstyle P}

![{\textstyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038590207af1024a629c1a08c855e9ac46bf5610) 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


{\textstyle 3:1}

![{\textstyle 3:1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdfd3570b8991fb33d61e324c67dd54f08a0eef), the longer segments being on the side of the vertex of the tetrahedron.
5. Find 



A
+
B


{\textstyle A+B}

![{\textstyle A+B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd957a1f1d7f480fb51c33b1e5ab3b8259cb37f) and 



2
A
−
3
B
+
I


{\textstyle 2A-3B+I}

![{\textstyle 2A-3B+I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9f21726daa3bc4c5b46cdebf17900d2384c4bc).
6. Find the products 



A
B


{\textstyle AB}

![{\textstyle AB}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac121a3e6c5dff2ebb2ff8340e4a7f1b35992e44) and 



B
A


{\textstyle BA}

![{\textstyle BA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d370a5b049f1bf605e0849e5011dab921a3e80) (and so make sure that, in general, 



A
B
≠
B
A


{\textstyle AB\neq BA}

![{\textstyle AB\neq BA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547dca1516d9c8402fe6b52671054d5412e39d7a) for matrices).
7. Find the inverse matrices for the given ones.
8. Find the determinants of the given matrices.
9. Point 



M


{\textstyle M}

![{\textstyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088) is the centroid of face 



B
C
D


{\textstyle BCD}

![{\textstyle BCD}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf276d48bdd841bc17d7b41806b4fefd4d93aec) of tetrahedron 



A
B
C
D


{\textstyle ABCD}

![{\textstyle ABCD}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56e9dd14b4fcb989c106f3679aa2699f07eee6d4). The old coordinate system is given by 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31), 






A
B

→




{\textstyle {\overrightarrow {AB}}}

![{\textstyle {\overrightarrow {AB}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9e6559e7cd7c16144340887587c73e294b4ba), 






A
C

→




{\textstyle {\overrightarrow {AC}}}

![{\textstyle {\overrightarrow {AC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cec90ca60916095907088cdf9c6807e322940609), 






A
D

→




{\textstyle {\overrightarrow {AD}}}

![{\textstyle {\overrightarrow {AD}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a09091c3ae3ae05dbb915b79e411ee15a84401f1), and the new coordinate system is given by 



M


{\textstyle M}

![{\textstyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913ace920108f7552777e36ac0b7ee3f5093a088), 






M
B

→




{\textstyle {\overrightarrow {MB}}}

![{\textstyle {\overrightarrow {MB}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc628c7e66ee539b78d720ffb8b08c68cda528bc), 






M
C

→




{\textstyle {\overrightarrow {MC}}}

![{\textstyle {\overrightarrow {MC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b29574c4ba08f6b03fc30de162aa30153d3ab0), 






M
A

→




{\textstyle {\overrightarrow {MA}}}

![{\textstyle {\overrightarrow {MA}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8be9dd7ec0faa12b4f2f9f335d65b1cab904c5). Find the coordinates of a point in the old coordinate system given its coordinates 




x
′



{\textstyle x'}

![{\textstyle x'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd4bbf7bc3c37c71d0a1f9e4ef6c504c8f9d5de), 




y
′



{\textstyle y'}

![{\textstyle y'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e40eea4dc71b01e5ddff1d14b9b89853fde81d), 




z
′



{\textstyle z'}

![{\textstyle z'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80546fbdafc04d89612c5491bc04d509709aeb17) in the new one.


#### Section 2


1. Two lines are given by the equations 





r


⋅


n


=
A


{\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}

![{\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bdfb3c03f3478452862b246cd0bec466eb624d) and 





r


=



r



0


+


a


t


{\textstyle {\textbf {r}}={\textbf {r}}\_{0}+{\textbf {a}}t}

![{\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca83af73cef59fe73380c9737197485304b371ec), and at that 





a


⋅


n


≠
0


{\textstyle {\textbf {a}}\cdot {\textbf {n}}\neq 0}

![{\textstyle {\textbf {a}}\cdot {\textbf {n}}\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33673ddea1dea180b68cac92d5065f32f85016b1). Find the position vector of the intersection point of these lines.
2. Find the distance from point 




M

0




{\textstyle M\_{0}}

![{\textstyle M_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feb4ebe89e459609fa9e97bf72ee561acb3f7836) with the position vector 






r



0




{\textstyle {\textbf {r}}\_{0}}

![{\textstyle {\textbf {r}}_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b1d2beb050ff53f1db5422ed3c6067091489d2) to the line defined by the equation (a) 





r


=



r



0


+


a


t


{\textstyle {\textbf {r}}={\textbf {r}}\_{0}+{\textbf {a}}t}

![{\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca83af73cef59fe73380c9737197485304b371ec); (b) 





r


⋅


n


=
A


{\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}

![{\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bdfb3c03f3478452862b246cd0bec466eb624d).
3. Diagonals of a rhombus intersect at point 



M
(
1
;

2
)


{\textstyle M(1;\,2)}

![{\textstyle M(1;\,2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5f919c98511e9602aff3b5c6fb6a44b3c5d75c), the longest of them being parallel to a horizontal axis. The side of the rhombus equals 2 and its obtuse angle is 




120

∘




{\textstyle 120^{\circ }}

![{\textstyle 120^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db4bd85704493ea3595bcef6a92dbadcef085e51). Compose the equations of the sides of this rhombus.
4. Compose the equations of lines passing through point 



A
(
2
;
−
4
)


{\textstyle A(2;-4)}

![{\textstyle A(2;-4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adb02900c49b8277ff0d9994f3c262b42712fc10) and forming angles of 




60

∘




{\textstyle 60^{\circ }}

![{\textstyle 60^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/880b547c0017493563c28efdad95fdac91f97352) with the line 






1
−
2
x

3


=



3
+
2
y


−
2





{\textstyle {\frac {1-2x}{3}}={\frac {3+2y}{-2}}}

![{\textstyle {\frac {1-2x}{3}}={\frac {3+2y}{-2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d3793ed8ac9c5b2a740ee251edbe89f3a25c7d4).
5. Find the cross product of (a) vectors 





a


(
3
;
−
2
;

1
)


{\textstyle {\textbf {a}}(3;-2;\,1)}

![{\textstyle {\textbf {a}}(3;-2;\,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24aa2e3d81f5e733ac961fc54c00c6400ac7c554) and 





b


(
2
;
−
5
;
−
3
)


{\textstyle {\textbf {b}}(2;-5;-3)}

![{\textstyle {\textbf {b}}(2;-5;-3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4f1d0f60888ec6d6e29a3ef51f7c26784b688e); (b) vectors 





a


(
3
;
−
2
;

1
)


{\textstyle {\textbf {a}}(3;-2;\,1)}

![{\textstyle {\textbf {a}}(3;-2;\,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24aa2e3d81f5e733ac961fc54c00c6400ac7c554) and 





c


(
−
18
;

12
;
−
6
)


{\textstyle {\textbf {c}}(-18;\,12;-6)}

![{\textstyle {\textbf {c}}(-18;\,12;-6)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/536804f73bf9994ca0ac6a28c7c18f9b7c025d00).
6. A triangle is constructed on vectors 





a


(
2
;
4
;
−
1
)


{\textstyle {\textbf {a}}(2;4;-1)}

![{\textstyle {\textbf {a}}(2;4;-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dcfef96a8fb429db141d89ad1a4a7d2b79796c7) and 





b


(
−
2
;
1
;
1
)


{\textstyle {\textbf {b}}(-2;1;1)}

![{\textstyle {\textbf {b}}(-2;1;1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33194e12e69d8cf8bef0e3a7a975b0bbc27e0d59). (a) Find the area of this triangle. (b) Find the altitudes of this triangle.
7. Find the scalar triple product of 





a


(
1
;

2
;
−
1
)


{\textstyle {\textbf {a}}(1;\,2;-1)}

![{\textstyle {\textbf {a}}(1;\,2;-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ff4c4819f08216a9b289eee65fae9e2409cd89), 





b


(
7
;
3
;
−
5
)


{\textstyle {\textbf {b}}(7;3;-5)}

![{\textstyle {\textbf {b}}(7;3;-5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad88c6efacea267911438883c60e91aa7e641d4), 





c


(
3
;

4
;
−
3
)


{\textstyle {\textbf {c}}(3;\,4;-3)}

![{\textstyle {\textbf {c}}(3;\,4;-3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3075cbf140d048ec7105863bf15ffbc46d4ea011).
8. It is known that basis vectors 






e



1




{\textstyle {\textbf {e}}\_{1}}

![{\textstyle {\textbf {e}}_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/633f38e385b970d94316cec9a3f0f8d8b2952c78), 






e



2




{\textstyle {\textbf {e}}\_{2}}

![{\textstyle {\textbf {e}}_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1141e813036efabb49e22e26699abda03db9c238), 






e



3




{\textstyle {\textbf {e}}\_{3}}

![{\textstyle {\textbf {e}}_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4136ef375267fae398133caf33d6309b3491fe75) have lengths of 



1


{\textstyle 1}

![{\textstyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6706df9ed9f240d1a94545fb4e522bda168fe8fd), 



2


{\textstyle 2}

![{\textstyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ed0cd8140e5a15b6fcce83602df58458e0f3b0), 



2


2




{\textstyle 2{\sqrt {2}}}

![{\textstyle 2{\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f302ae86c5e346beaf1dc60ab28ce51583d3a10f) respectively, and 



∠
(



e



1


,



e



2


)
=

120

∘




{\textstyle \angle ({\textbf {e}}\_{1},{\textbf {e}}\_{2})=120^{\circ }}

![{\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{2})=120^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc355fb33e21d27781b3c69f604dfa4ac8a2335), 



∠
(



e



1


,



e



3


)
=

135

∘




{\textstyle \angle ({\textbf {e}}\_{1},{\textbf {e}}\_{3})=135^{\circ }}

![{\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{3})=135^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/077d554778a6237a758bdd3cd174fb71e56c5a59), 



∠
(



e



2


,



e



3


)
=

45

∘




{\textstyle \angle ({\textbf {e}}\_{2},{\textbf {e}}\_{3})=45^{\circ }}

![{\textstyle \angle ({\textbf {e}}_{2},{\textbf {e}}_{3})=45^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c625e201eaf56a5ab2f55c468fb74919eecf950). Find the volume of a parallelepiped constructed on vectors with coordinates 



(
−
1
;

0
;

2
)


{\textstyle (-1;\,0;\,2)}

![{\textstyle (-1;\,0;\,2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f80a6dd17132920d07de233c927f5b35fe55342), 



(
1
;

1

4
)


{\textstyle (1;\,1\,4)}

![{\textstyle (1;\,1\,4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93f2f019128f43ccd8b9bf1ec5fae847335c9609) and 



(
−
2
;

1
;

1
)


{\textstyle (-2;\,1;\,1)}

![{\textstyle (-2;\,1;\,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c29df73c5099969e89def9536b89bed1f8b8300b) in this basis.


#### Section 3


1. Prove that a curve given by 



34

x

2


+
24
x
y
+
41

y

2


−
44
x
+
58
y
+
1
=
0


{\textstyle 34x^{2}+24xy+41y^{2}-44x+58y+1=0}

![{\textstyle 34x^{2}+24xy+41y^{2}-44x+58y+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2081edc20b56611eb9f1897a625f3bc0bc3df3b5) 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) 



9

x

2


−
16

y

2


−
6
x
+
8
y
−
144
=
0


{\textstyle 9x^{2}-16y^{2}-6x+8y-144=0}

![{\textstyle 9x^{2}-16y^{2}-6x+8y-144=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2307372c3cf01eef66761c99845dcbe394be29f); (b) 



9

x

2


+
4

y

2


+
6
x
−
4
y
−
2
=
0


{\textstyle 9x^{2}+4y^{2}+6x-4y-2=0}

![{\textstyle 9x^{2}+4y^{2}+6x-4y-2=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73b831469fdf03eac9e9c9656ae7fef80096fb94); (c) 



12

x

2


−
12
x
−
32
y
−
29
=
0


{\textstyle 12x^{2}-12x-32y-29=0}

![{\textstyle 12x^{2}-12x-32y-29=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cde74f6df39c7d1f475a31c4f1add253f1a43eb); (d) 



x
y
+
2
x
+
y
=
0


{\textstyle xy+2x+y=0}

![{\textstyle xy+2x+y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ced2a1c13f7eb77948ffc6ba9c1789659873573);
3. Find the equations of lines tangent to curve 



6
x
y
+
8

y

2


−
12
x
−
26
y
+
11
=
0


{\textstyle 6xy+8y^{2}-12x-26y+11=0}

![{\textstyle 6xy+8y^{2}-12x-26y+11=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38463f126baaf801dce5a01c4e5d2e4c9d5e5080) that are (a) parallel to line 



6
x
+
17
y
−
4
=
0


{\textstyle 6x+17y-4=0}

![{\textstyle 6x+17y-4=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1136341a1ea0ad8bbc7101dbceadb86b33e90e); (b) perpendicular to line 



41
x
−
24
y
+
3
=
0


{\textstyle 41x-24y+3=0}

![{\textstyle 41x-24y+3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91928594589b3b61e71e2f6d60125360b48cd4fd); (c) parallel to line 



y
=
2


{\textstyle y=2}

![{\textstyle y=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/887628e811c56cc94aaccb0f5e4c773f19231cf9).
4. For each value of parameter 



a


{\textstyle a}

![{\textstyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8) determine types of surfaces given by the equations: (a) 




x

2


+

y

2


−

z

2


=
a


{\textstyle x^{2}+y^{2}-z^{2}=a}

![{\textstyle x^{2}+y^{2}-z^{2}=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aedac9a12be3e2a8bb6549e080cecd007665a44c); (b) 




x

2


+
a

(


y

2


+

z

2



)

=
1


{\textstyle x^{2}+a\left(y^{2}+z^{2}\right)=1}

![{\textstyle x^{2}+a\left(y^{2}+z^{2}\right)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04136582473b57e7f0525feeebccc4460ecf7ad8); (c) 




x

2


+
a

y

2


=
a
z


{\textstyle x^{2}+ay^{2}=az}

![{\textstyle x^{2}+ay^{2}=az}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9326d877d128ad9d17a96c9765e1da5f7c91e6); (d) 




x

2


+
a

y

2


=
a
z
+
1


{\textstyle x^{2}+ay^{2}=az+1}

![{\textstyle x^{2}+ay^{2}=az+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1cd37bf6d43b7adafdd4ec58f392dd620872e3).
5. Find a vector equation of a right circular cone with apex 




M

0



(



r



0


)



{\textstyle M\_{0}\left({\textbf {r}}\_{0}\right)}

![{\textstyle M_{0}\left({\textbf {r}}_{0}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b67abdb7e98da13a4de428625f69d249276ae5) and axis 





r


=



r



0


+


a


t


{\textstyle {\textbf {r}}={\textbf {r}}\_{0}+{\textbf {a}}t}

![{\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca83af73cef59fe73380c9737197485304b371ec) if it is known that generatrices of this cone form the angle of 



α


{\textstyle \alpha }

![{\textstyle \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185) with its axis.
6. Find the equation of a cylinder with radius 





2




{\textstyle {\sqrt {2}}}

![{\textstyle {\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5094a5b1e2f42490aa4de2c7a4b7235a27f1b73f) that has an axis 



x
=
1
+
t


{\textstyle x=1+t}

![{\textstyle x=1+t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca54a99a9fdad0ee1a0bba54ac67014e83a51e2), 



y
=
2
+
t


{\textstyle y=2+t}

![{\textstyle y=2+t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/213ba8ad271ceb3b25aaaafcbe23edd7c885c7f0), 



z
=
3
+
t


{\textstyle z=3+t}

![{\textstyle z=3+t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/582781cac22a2c157f5b5b60da347600c9778a6f).
7. An ellipsoid is symmetric with respect to coordinate planes, passes through point 



M
(
3
;

1
;

1
)


{\textstyle M(3;\,1;\,1)}

![{\textstyle M(3;\,1;\,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6192e6c5728edd5b9a22b9c19aa8be852e2f0599) and circle 




x

2


+

y

2


+

z

2


=
9


{\textstyle x^{2}+y^{2}+z^{2}=9}

![{\textstyle x^{2}+y^{2}+z^{2}=9}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29376da667d60dba7e1f552a56d1ce98caea3a4d), 



x
−
z
=
0


{\textstyle x-z=0}

![{\textstyle x-z=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/709115599ad054688b92e60cdb84a4f32fe944f1). Find the equation of this ellipsoid.


### The retake exam


Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.