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REVIEW OF MATHEMATICS

REVIEW OF MATHEMATICS. Given. Dot product:.  is the angle between the two vectors. Example:. Magnitude of vector:. Example:. Review of Vectors Analysis. Review of Vectors Analysis. Vectors and are said to be perpendicular or orthogonal if. Example:.

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REVIEW OF MATHEMATICS

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  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

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