Linear algebra is a fundamental component of continuum mechanics (and most engineering disciplines). Solve the following matrix and vector problems showing all steps for full credit. As it is always a good idea to be able to find and correct your own mistakes, you are encouraged to use MATLAB, Maple, Mathematica, etc. to check your answers, but you still need to show all step.
a. Determine if the vectors [u] = [7 3 -2] and [v] = [2 -4 -1] are orthogonal to each other.
b. A plane is described by the vectors [u] = [3 -9 2] and [v] = [11 2 -6]. Find a vector that is normal to this plane.
c. Find the tensor (outer) product of the vectors [u] = [4 -2 3] and [v] = [1 8 -3] (i.e., u v).
d. A point in a material is represented by the position vector X in the reference configuration. The material undergoes a 75 degree counter-clockwise rotation about the z-axis. Find the coordinates of the material point x after the rotation. [X] = [2 3 0], [F] = [cos theta sin theta 0 -sin theta cos theta 0 0 0 1], x = F middot x
e. Confirm that the angle between X and x is indeed 75 degree.