Page 85 - Introduction to Continuum Mechanics

70 Problems

2B3. A tensor T transforms the base vectors *i and e 2 so that

2B4. Obtain the matrix for the tensor T which transforms the base vectors as follows:

2B5. Find the matrk of the tensor T which transforms any vector a into a vector b = nt(a -n)
where

2B6. (a) A tensor T transforms every vector into its mirror image with respect to the plane
whose normal is e 2. Find the matrix of T.
b) Do part (a) if the plane has a normal in the 63 direction instead.
2B7. a) Let R correspond to a right-hand rotation of angle 6 about the jq-axis. Find the matrk
ofR.
b) Do part (a) if the rotation is about the *2-axis.

2B8. Consider a plane of reflection which passes through the origin. Let n be a unit normal
vector to the plane and let r be the position vector for a point in space
(a) Show that the reflected vector for r is given by Tr= r-2(r-n)n, where T is the
transformation that corresponds to the reflection.

(b) Let n=^73'(ei+e2+e3), find the matrk of the linear transformation T that corresponds to
this reflection.
(c) Use this linear transformation to find the mirror image of a vector a = ej+262+303.
2B9. A rigid body undergoes a right hand rotation of angle 0 about an axis which is in the
direction of the unit vector m. Let the origin of the coordinates be on the axis of rotation and
r be the position vector for a typical point in the body.
(a) Show that the rotated vector of r is given by Rr= (l-cos0)(mT)m+cos0r4-sm0inXr,
where R is the transformation that corresponds to the rotation.