3. Transformation of vector components

Try these first

1. 1.

   Define a rotation matrix and its role in vector transformation.

   Hint

   Think about how angles affect the orientation of the vector.

2. 2.

   What do we multiply vector components by to transform their representation?

   Hint

   What version of the rotation matrix are we using?

3. 3.

   What must you do to transform vector components from one coordinate system to another?

   • Multiply by the rotation matrix
   • Multiply by the transpose of the rotation matrix
   • Add both systems' components

   Hint

   Which matrix is applied to the components?

4. 4.

   Is it true that the vector itself remains unchanged during transformation?

   • True
   • False

   Hint

   Think about how you express the same vector in different systems.

5. 5.

   A force vector is represented as (10 N, 0 N) along the x-axis. If the system is rotated 90 degrees counterclockwise, what will its new components be?

   Hint

   Think about what a 90-degree rotation does to the orientation of the axes!

6. 6.

   You have a vector expressed as (2, -3) in one coordinate system. The second coordinate system is rotated 60 degrees clockwise. What are the new coordinates?

   Hint

   Remember to use co-functions for the angles when computing trigonometric values.