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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']

Course Topics

Course Sections and Topics

Section	Topics within the section
Vector algebra	Vector spaces Basic operations on vectors (summation, multiplication by scalar, dot product) Linear dependency and in-dependency of the vectors Basis in vector spaces
Introduction to matrices and determinants	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
Lines in the plane and in the space	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
Planes in the space	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
Quadratic curves	Circle Ellipse Hyperbola Parabola Canonical equations Shifting of coordinate system Rotating of coordinate system Parametrization
Quadric surfaces	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

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

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 ${\textstyle |{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}}$ given that ${\textstyle |{\textbf {a}}|=4}$ , ${\textstyle |{\textbf {b}}|=1}$ , ${\textstyle \angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }}$	0
Question	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	0
Question	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	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 ${\textstyle 3:1}$ , 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 ${\textstyle A}$ 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 ${\textstyle ABC}$ ?	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 ${\textstyle A+B}$ and ${\textstyle 2A-3B+I}$	0
Question	Find the products ${\textstyle AB}$ and ${\textstyle BA}$ (and so make sure that, in general, ${\textstyle AB\neq BA}$ for matrices)	0
Question	Find the inverse matrices for the given ones	0
Question	Find the determinants of the given matrices	0
Question	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	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 ${\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}$ 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	0
Question	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) ${\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}$	0
Question	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	0
Question	Compose the equations of lines passing through point ${\textstyle A(2;-4)}$ and forming angles of ${\textstyle 60^{\circ }}$ with the line ${\textstyle {\frac {1-2x}{3}}={\frac {3+2y}{-2}}}$	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 ${\textstyle {\textbf {a}}(3;-2;\,1)}$ and ${\textstyle {\textbf {b}}(2;-5;-3)}$ ; (b) vectors ${\textstyle {\textbf {a}}(3;-2;\,1)}$ and ${\textstyle {\textbf {c}}(-18;\,12;-6)}$	0
Question	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	0
Question	Find the scalar triple product of ${\textstyle {\textbf {a}}(1;\,2;-1)}$ , ${\textstyle {\textbf {b}}(7;3;-5)}$ , ${\textstyle {\textbf {c}}(3;\,4;-3)}$	0
Question	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	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 ${\textstyle 34x^{2}+24xy+41y^{2}-44x+58y+1=0}$ 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	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) ${\textstyle 9x^{2}-16y^{2}-6x+8y-144=0}$ ; (b) ${\textstyle 9x^{2}+4y^{2}+6x-4y-2=0}$ ; (c) ${\textstyle 12x^{2}-12x-32y-29=0}$ ; (d) ${\textstyle xy+2x+y=0}$ ;	0
Question	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 ${\textstyle y=2}$	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 ${\textstyle a}$ 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) ${\textstyle x^{2}+ay^{2}=az+1}$	0
Question	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	0
Question	Find the equation of a cylinder with radius ${\textstyle {\sqrt {2}}}$ that has an axis ${\textstyle x=1+t}$ , ${\textstyle y=2+t}$ , ${\textstyle z=3+t}$	0
Question	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	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