Course Overview: The course is an introduction to some elementary ideas in the geometry of euclidean space through vectors. One focus of the course is the use of co-ordinates and an appreciation of the invariance of geometry under an orthogonal change of variable. This leads into a deeper study of orthogonal matrices, of rotating frames, and into related co-ordinate systems.
Course Syllabus:
See the examinable syllabus.
Lecturer(s):
Dr Richard Earl
Learning Outcomes: Students will learn how to encode a geometric scenario into vector equations and meet the vector algebra needed to manipulate such equations. Students will meet the benefits of choosing sensible co-ordinate systems and appreciate what geometry is invariant of such choices.
Course Synopsis: Euclidean geometry in two and three dimensions approached by vectors and coordinates. Vector addition and scalar multiplication. The scalar product, equations of planes, lines and circles. [3]
The vector product in three dimensions. Use of \(\mathbf{a}, \mathbf{b}, \mathbf{a} \land \mathbf{b}\) as a basis. \(\mathbf{r} \land \mathbf{a} = \mathbf{b}\) represents a line. Scalar triple products and vector triple products, vector algebra. [2]
Conics (normal form only), focus and directrix. Showing the locus \(Ax^2 + Bxy + Cy^2 = 1\) can be put in normal form via a rotation matrix. Orthogonal matrices. \(2\times 2\) orthogonal matrices and the maps they represent. Orthonormal
bases in \(\mathbb{R}^3\). Orthogonal change of variable; \(A\mathbf{u} \cdot A\mathbf{v} = \mathbf{u \cdot v}\) and \(A(\mathbf{u} \land \mathbf{v}) = \pm A\mathbf{u} \land A \mathbf{v}\). Statement that a real symmetric matrix can be orthogonally diagonalized. Simple examples identifying conics not in normal form. [3]
\(3 \times 3\) orthogonal matrices; \(SO(3)\) and rotations; conditions for being a reflection. Isometries of \(\mathbb{R}^3\). [2]
Rotating frames in \(2\) and \(3\) dimensions. Angular velocity. \(\mathbf{v} = \omega \land \mathbf{r}\). [1]
Parametrised surfaces, including spheres, cones. Examples of coordinate systems including parabolic, spherical and cylindrical polars. Calculating normal as \(\mathbf{r}_u \land \mathbf{r}_v\). Surface area. [4]
