1 / 20

REVIEW OF MATHEMATICS

This review covers dot product, cross product, vector triple product, scalar triple product, eigenvalues, and eigenvectors analysis with examples and solutions.

monicanoland
Download Presentation

REVIEW OF MATHEMATICS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. REVIEW OF MATHEMATICS

  2. Given Dot product:  is the angle between the two vectors. Example: Magnitude of vector: Example: Review of Vectors Analysis

  3. Review of Vectors Analysis Vectors and are said to be perpendicular or orthogonal if Example: Note that the above vectors represent the unit vectors for the X-axis and Y-axis. They are definitely perpendicular or orthogonal.

  4. Cross product: • is the angle between the two vectors. Example: Review of Vectors Analysis

  5. Example: Review of Vectors Analysis The cross product of and provides us with a vector which is perpendicular to both and Note that the above vectors represent the unit vectors for the X-axis and Y-axis respectively. Their cross product is the unit vector for the Z-axis, which is definitely perpendicular to both the X-axis and the Y-axis.

  6. Review of Vectors Analysis Note that the unit vectors for the right handed Cartesian reference frame are orthonormal basis vectors, i.e.

  7. Vector triple product: Review of Vectors Analysis Example:

  8. Example: Review of Vectors Analysis Scalar triple product:

  9. Given Example: Given where  is a any constant Example: Review of Vectors Analysis

  10. Given Example: Review of Vectors Analysis

  11. Given Example: Review of Vectors Analysis

  12. Example: Review of Vectors Analysis Given where A is a matrix of dimension comparable to the vector being multiplied

  13. Let A be an nn matrix. If there exists a  and a nonzero n1 vector such that then  is called an eigenvalue of A and is called an eigenvector of A corresponding to the eigenvalue  Eigenvalues and Eigenvectors Let In be a nn identity matrix. The eigenvalues of nn matrix A can be obtained from: A nn matrix A has at least one and at most “n” distinct eigenvalues

  14. Solution: Example 1: Eigenvalues and Eigenvectors Find the eigenvalues of

  15. at =1? What is the eigenvector of Example 2: Eigenvalues and Eigenvectors

  16. Example 2: Eigenvalues and Eigenvectors Multiply 3rd eqn by -5 and add it to 1st eqn to eliminate

  17. Divide 2nd eqn by and simplify using the known result: Example 2: Eigenvalues and Eigenvectors

  18. Example 2: Eigenvalues and Eigenvectors Story so far: We can obtain a normalized eigenvector using:

  19. Trigonometric Functions

  20. Trigonometric Functions

More Related