5. Rotation tensor

Try these first

1. 1.

   Define a rotation tensor in your own words.

   Hint

   Think about the physical meaning of rotating an object.

2. 2.

   What is the determinant of a rotation matrix?

   Hint

   Consider how volume changes during rotation.

3. 3.

   What is a fundamental property of rotation tensors?

   • They alter vector magnitudes
   • They preserve vector magnitudes
   • They only operate in two dimensions

   Hint

   Consider what happens when you spin an object.

4. 4.

   True or False: The determinant of a rotation matrix is less than one.

   • True
   • False

   Hint

   Reflect on the conservation of volume during rotations.

5. 5.

   If a vector (2, 0, 0) is rotated 90 degrees around the z-axis, what will be its new coordinates?

   Hint

   Use the rotation matrix: R = [[0, -1, 0], [1, 0, 0], [0, 0, 1]].

6. 6.

   Consider a 3D object being rotated in a simulation. How can rotation tensors ensure that its size remains consistent?

   Hint

   Reflect on the properties of orthonormal matrices.