Question

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.

Answer 1

A ) Not orthogonal to each other

B) 50i + 40j + 105k

C) The tensor product is attached below

D ) The value of X = F.X is attached below

Explanation:

attached below is the detailed solution of the above problem

A) for the vectors ( u ) and ( v ) to be orthogonal to each other [ U.V has to be = 0 ] but in this scenario U.V = 4 hence they are not orthogonal to each other

b) The vector normal to plane is gotten by : U x V

= 50i + 40j + 105k