BSc: Analytic Geometry And Linear Algebra I.f22 - Revision history

I.konyukhov: /* Course activities and grading breakdown */

2023-08-31T16:00:55Z

Course activities and grading breakdown

← Older revision		Revision as of 16:00, 31 August 2023
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	\| Tests\|\| 30 (15 for each)		\| Tests\|\| 30 (15 for each)
	\|-		\|-
−	\| Final exam \|\| 30	+	\| Final exam \|\| 35
−	\|-
−	\| In-class participation \|\| 5 extras
	\|}		\|}

V.ivanov at 10:22, 29 August 2022

2022-08-29T10:22:03Z

← Older revision		Revision as of 10:22, 29 August 2022
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	! Activity Type !! Percentage of the overall course grade		! Activity Type !! Percentage of the overall course grade
	\|-		\|-
−	\| Midterm \|\| 30	+	\| Midterm \|\| 35
	\|-		\|-
−	\| Tests\|\| 20 (10 for each)	+	\| Tests\|\| 30 (15 for each)
	\|-		\|-
−	\| Final exam \|\| 50	+	\| Final exam \|\| 30
	\|-		\|-
	\| In-class participation \|\| 5 extras		\| In-class participation \|\| 5 extras

I.konyukhov: /* Final assessment */

2022-06-28T08:14:04Z

Final assessment

← Older revision		Revision as of 08:14, 28 June 2022
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	=== Final assessment ===		=== Final assessment ===
−	~~'''~~Section 1~~'''~~	+	==== Section 1 ====
	# Evaluate <math display="inline">\|\textbf{a}\|^2-2\sqrt3\textbf{a}\cdot\textbf{b}-7\|\textbf{b}\|^2</math> given that <math display="inline">\|\textbf{a}\|=4</math>, <math display="inline">\|\textbf{b}\|=1</math>, <math display="inline">\angle(\textbf{a},\,\textbf{b})=150^{\circ}</math>.		# Evaluate <math display="inline">\|\textbf{a}\|^2-2\sqrt3\textbf{a}\cdot\textbf{b}-7\|\textbf{b}\|^2</math> given that <math display="inline">\|\textbf{a}\|=4</math>, <math display="inline">\|\textbf{b}\|=1</math>, <math display="inline">\angle(\textbf{a},\,\textbf{b})=150^{\circ}</math>.
	# Prove that vectors <math display="inline">\textbf{b}(\textbf{a}\cdot\textbf{c})-\textbf{c}(\textbf{a}\cdot\textbf{b})</math> and <math display="inline">\textbf{a}</math> are perpendicular to each other.		# Prove that vectors <math display="inline">\textbf{b}(\textbf{a}\cdot\textbf{c})-\textbf{c}(\textbf{a}\cdot\textbf{b})</math> and <math display="inline">\textbf{a}</math> are perpendicular to each other.
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	# Point <math display="inline">M</math> is the centroid of face <math display="inline">BCD</math> of tetrahedron <math display="inline">ABCD</math>. The old coordinate system is given by <math display="inline">A</math>, <math display="inline">\overrightarrow{AB}</math>, <math display="inline">\overrightarrow{AC}</math>, <math display="inline">\overrightarrow{AD}</math>, and the new coordinate system is given by <math display="inline">M</math>, <math display="inline">\overrightarrow{MB}</math>, <math display="inline">\overrightarrow{MC}</math>, <math display="inline">\overrightarrow{MA}</math>. Find the coordinates of a point in the old coordinate system given its coordinates <math display="inline">x'</math>, <math display="inline">y'</math>, <math display="inline">z'</math> in the new one.		# Point <math display="inline">M</math> is the centroid of face <math display="inline">BCD</math> of tetrahedron <math display="inline">ABCD</math>. The old coordinate system is given by <math display="inline">A</math>, <math display="inline">\overrightarrow{AB}</math>, <math display="inline">\overrightarrow{AC}</math>, <math display="inline">\overrightarrow{AD}</math>, and the new coordinate system is given by <math display="inline">M</math>, <math display="inline">\overrightarrow{MB}</math>, <math display="inline">\overrightarrow{MC}</math>, <math display="inline">\overrightarrow{MA}</math>. Find the coordinates of a point in the old coordinate system given its coordinates <math display="inline">x'</math>, <math display="inline">y'</math>, <math display="inline">z'</math> in the new one.

−	~~'''~~Section 2~~'''~~	+	==== Section 2 ====
	# Two lines are given by the equations <math display="inline">\textbf{r}\cdot\textbf{n}=A</math> and <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>, and at that <math display="inline">\textbf{a}\cdot\textbf{n}\neq0</math>. Find the position vector of the intersection point of these lines.		# Two lines are given by the equations <math display="inline">\textbf{r}\cdot\textbf{n}=A</math> and <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>, and at that <math display="inline">\textbf{a}\cdot\textbf{n}\neq0</math>. Find the position vector of the intersection point of these lines.
	# Find the distance from point <math display="inline">M_0</math> with the position vector <math display="inline">\textbf{r}_0</math> to the line defined by the equation (a) <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>; (b) <math display="inline">\textbf{r}\cdot\textbf{n}=A</math>.		# Find the distance from point <math display="inline">M_0</math> with the position vector <math display="inline">\textbf{r}_0</math> to the line defined by the equation (a) <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>; (b) <math display="inline">\textbf{r}\cdot\textbf{n}=A</math>.
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	# It is known that basis vectors <math display="inline">\textbf{e}_1</math>, <math display="inline">\textbf{e}_2</math>, <math display="inline">\textbf{e}_3</math> have lengths of <math display="inline">1</math>, <math display="inline">2</math>, <math display="inline">2\sqrt2</math> respectively, and <math display="inline">\angle(\textbf{e}_1,\textbf{e}_2)=120^{\circ}</math>, <math display="inline">\angle(\textbf{e}_1,\textbf{e}_3)=135^{\circ}</math>, <math display="inline">\angle(\textbf{e}_2,\textbf{e}_3)=45^{\circ}</math>. Find the volume of a parallelepiped constructed on vectors with coordinates <math display="inline">(-1;\,0;\,2)</math>, <math display="inline">(1;\,1\,4)</math> and <math display="inline">(-2;\,1;\,1)</math> in this basis.		# It is known that basis vectors <math display="inline">\textbf{e}_1</math>, <math display="inline">\textbf{e}_2</math>, <math display="inline">\textbf{e}_3</math> have lengths of <math display="inline">1</math>, <math display="inline">2</math>, <math display="inline">2\sqrt2</math> respectively, and <math display="inline">\angle(\textbf{e}_1,\textbf{e}_2)=120^{\circ}</math>, <math display="inline">\angle(\textbf{e}_1,\textbf{e}_3)=135^{\circ}</math>, <math display="inline">\angle(\textbf{e}_2,\textbf{e}_3)=45^{\circ}</math>. Find the volume of a parallelepiped constructed on vectors with coordinates <math display="inline">(-1;\,0;\,2)</math>, <math display="inline">(1;\,1\,4)</math> and <math display="inline">(-2;\,1;\,1)</math> in this basis.

−	~~'''~~Section 3~~'''~~	+	==== Section 3 ====
	# Prove that a curve given by <math display="inline">34x^2+24xy+41y^2-44x+58y+1=0</math> 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.		# Prove that a curve given by <math display="inline">34x^2+24xy+41y^2-44x+58y+1=0</math> 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.
	# 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) <math display="inline">9x^2-16y^2-6x+8y-144=0</math>; (b) <math display="inline">9x^2+4y^2+6x-4y-2=0</math>; (c) <math display="inline">12x^2-12x-32y-29=0</math>; (d) <math display="inline">xy+2x+y=0</math>;		# 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) <math display="inline">9x^2-16y^2-6x+8y-144=0</math>; (b) <math display="inline">9x^2+4y^2+6x-4y-2=0</math>; (c) <math display="inline">12x^2-12x-32y-29=0</math>; (d) <math display="inline">xy+2x+y=0</math>;

I.konyukhov: /* Resources, literature and reference materials */

2022-06-23T21:02:24Z

Resources, literature and reference materials

← Older revision		Revision as of 21:02, 23 June 2022
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	* V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. Textbook. Tomsk: TPU Press, 2009, 114 pp [https://portal.tpu.ru/SHARED/k/KONVAL/Textbooks/Tab1/Konev-Linear_Algebra_Vector_Algebra_and_Analytical_Geome.pdf book1]		* V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. Textbook. Tomsk: TPU Press, 2009, 114 pp [https://portal.tpu.ru/SHARED/k/KONVAL/Textbooks/Tab1/Konev-Linear_Algebra_Vector_Algebra_and_Analytical_Geome.pdf book1]
	* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [https://arxiv.org/pdf/1111.6521.pdf book2]		* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [https://arxiv.org/pdf/1111.6521.pdf book2]
−	* P.R. Vital. Analytical ~~Geometru~~ 2D and 3D Analytical Geometry 2D and 3D [https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604 book3]	+	* P.R. Vital. Analytical Geometry 2D and 3D Analytical Geometry 2D and 3D [https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604 book3]

	== Activities and Teaching Methods ==		== Activities and Teaching Methods ==

I.konyukhov: /* Resources, literature and reference materials */

2022-06-23T21:02:09Z

Resources, literature and reference materials

← Older revision		Revision as of 21:02, 23 June 2022
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	* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [https://arxiv.org/pdf/1111.6521.pdf book2]		* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [https://arxiv.org/pdf/1111.6521.pdf book2]
	* P.R. Vital. Analytical Geometru 2D and 3D Analytical Geometry 2D and 3D [https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604 book3]		* P.R. Vital. Analytical Geometru 2D and 3D Analytical Geometry 2D and 3D [https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604 book3]
−
−
−	=== Software and tools used within the course ===
−	* No.

	== Activities and Teaching Methods ==		== Activities and Teaching Methods ==

I.konyukhov: /* Analytical Geometry & Linear Algebra – I */

2022-06-23T20:49:16Z

Analytical Geometry & Linear Algebra – I

← Older revision		Revision as of 20:49, 23 June 2022
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	* '''Course name''': Analytical Geometry & Linear Algebra – I		* '''Course name''': Analytical Geometry & Linear Algebra – I
	* '''Code discipline''': CSE202		* '''Code discipline''': CSE202
−	* '''Subject area''': Math~~. Computer Science~~	+	* '''Subject area''': Math
	== Short Description ==		== 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.		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.

I.konyukhov: /* Final assessment */

2022-06-23T20:46:12Z

Final assessment

@@ Line 230: / Line 230: @@
 === Final assessment ===
 '''Section 1'''
 '''Section 2'''
 '''Section 3'''
 === The retake exam ===

I.konyukhov: /* Section 3 */

2022-06-23T20:44:19Z

Section 3

@@ Line 219: / Line 219: @@
 ==== Section 3 ====
-#What is the difference between undirected and directed graphs?
+# What is the type of a quadric surface given by a certain equation?
 === Final assessment ===

I.konyukhov: /* Section 2 */

2022-06-23T20:43:47Z

Section 2

← Older revision		Revision as of 20:43, 23 June 2022
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	==== Section 2 ====		==== Section 2 ====
		+	# How to represent a line in the vector form?
−
		+	# What is the result of intersection of two planes in vector form?
		+	# How to derive the formula for the distance from a point to a line?
		+	# How to interpret geometrically the distance between lines?
		+	# List all possible inter-positions of lines in the space.
		+	# What is the difference between general and normalized forms of equations of a plane?
		+	# How to rewrite the equation of a plane in a vector form?
		+	# What is the normal to a plane?
		+	# How to interpret the cross products of two vectors?
		+	# What is the meaning of scalar triple product of three vectors?

	==== Section 3 ====		==== Section 3 ====

I.konyukhov: /* Section 1 */

2022-06-23T20:43:05Z

Section 1

← Older revision		Revision as of 20:43, 23 June 2022
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	==== Section 1 ====		==== Section 1 ====
		+	# How to perform the shift of the vector?
−	#Solve Truth Tables
		+	# What is the geometrical interpretation of the dot product?
−	#Use Truth Tables to analyse arguments
		+	# How to determine whether the vectors are linearly dependent?
−	#Use Quantifiers to assess inferences
−	#What is ~~Propositional Logic~~ ~~used~~ ~~for~~?	+	# What is a vector basis?
−	#What is ~~Predicate~~ ~~Logic~~ ~~used~~ ~~for~~?	+	# What is the difference between matrices and determinants?
		+	# Matrices <math display="inline">A</math> and <math display="inline">C</math> have dimensions of <math display="inline">m\times n</math> and <math display="inline">p\times q</math> respectively, and it is known that the product <math display="inline">ABC</math> exists. What are possible dimensions of <math display="inline">B</math> and <math display="inline">ABC</math>?
		+	# How to determine the rank of a matrix?
		+	# What is the meaning of the inverse matrix?
		+	# How to restate a system of linear equations in the matrix form?

	==== Section 2 ====		==== Section 2 ====