REVIEW OF MATHEMATICS Review of Vectors Analysis Given

REVIEW OF MATHEMATICS

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

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

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

Review of Vectors Analysis The cross product of and which is perpendicular to both provides us with a vector and Example: Note that the above vectors represent the unit vectors for the Xaxis 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.

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

Review of Vectors Analysis Vector triple product: Example:

Review of Vectors Analysis Scalar triple product: Example:

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

Review of Vectors Analysis Given Example:

Review of Vectors Analysis Given Example:

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

Eigenvalues and Eigenvectors 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 of A corresponding to the eigenvalue is called an eigenvector 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

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

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

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

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

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

Trigonometric Functions

Trigonometric Functions