MAT 220: Vector Geometry and Algebra

Stony Brook University - Spring 2017

About the course

The course covers the following topics:
Vectors and vector algebra. Dot product. Cross product and triple product. Complex numbers and quaternions and their geometric interpretations. symmetry, similarity transformations, affine transformations on a plane and n 3-space. Analytical geometry. Vector equations of lines and planes. Curves and surfaces of degree two. The basic notions of projective geometry. No preliminary knowledge of advanced mathematics is required.


Level 4 on the mathematics placement examination, or permission by instructor.

Lectures time and location

Tuesdays and Thursdays 4:00pm - 5:20pm in Physics P112


Oleg Viro
Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University
Arrived at Stony Brook in 2007.

Office: Math Tower 5-110
Phone: (631) 632-8286
Email: oleg.viro AT
Web page:

Research fields: Topology and Geometry,
especially low-dimensional topology and real algebraic geometry.

Office hours

Tuesdays and Thursdays 5:40pm - 6:50pm in Math Tower 5-110.


Patrick Tonra,
Arrived at Stony Brook in 2004
Office: Math Tower 4-122
Email: jean-francois.arbour AT


Homework sets will be typically assigned weekly and due on Thursdays in class. They will be posted on the blackboard site. Late homework will not be accepted. However, grades for homework assignments may be dropped in cases of documented medical problems or similar difficulties.


There will be two in-class midterms; the first midterm will be on Th, 3/9. The date of the second midterm will be announced later.

Final Exam: Monday, May 15, 2:15pm-5:00pm

Grading policy

Grades will be based on the following scheme: Homework -- 10%; Quizes -- 10%, Midterms -- 20% each; Final Exam 40%.

Program of the course

Complex numbers. The basic arithmetic of complex numbers: addition, subtraction, multiplication, conjugation, division, moduli, absolute value and argument, trigonometric form of a complex number. Geometry of multiplication.

Quaternions. Quaternion units, scalars and vectors, multiplication, conjugation, division, absolute value, unit quaternions.

Vectors and vector operations. The basic operations with vectors. Linear dependence. Bases and dimension. Arithmetic vector spaces $\mathbb R^n$. Applications to geometric problems.

Multiplications: dot and cross products. Algebraic properties and geometry of them, distance and dot product, projections to lines and planes, volume and triple product.

Analytic geometry of lines and planes. Line on a plane, their vector and coordinate equations. Various types of equations. Formulas for distances.

Transformations: isometries. Types of isometries of plane and 3-space. Their classification. Applications to geometric problems.

Similarity transformations. Classification. Relation to complex numbers.

Affine transformations. Affine invariants. Center of masses.

Polyhedra and their plane sections.

Conic sections. Ellipse, parabola, hyperbola. Equations, focal and directorial properties.

Quadratic surfaces. Cylinders, cones, ellipsoids, paraboloids, hyperboloids.

Projective spaces

Projective lines. Double ratio.

Projective geometry. Duality.


If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the Evacuation Guide for People with Physical Disabilities.

Academic Integrity

Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website.

Critical Incident Management

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.