26.13. Orthogonality and Orthonormal Sets

Try these first

1. 1.

   Define orthogonality in your own words.

   Hint

   Think about the angle between the vectors.

2. 2.

   What do we call a set of vectors that are all orthogonal and have unit length?

   Hint

   Starts with 'o'.

3. 3.

   What does it mean for two vectors to be orthogonal?

   • They have the same direction
   • Their inner product is zero
   • They are parallel

   Hint

   Think about the angles between them.

4. 4.

   True or False: An orthonormal set can consist of vectors that are not unit vectors.

   • True
   • False

   Hint

   Recall the definition of orthonormality.

5. 5.

   Given the vectors u = (1, 2) and v = (3, 4), determine if they are orthogonal and convert them into an orthonormal set.

   Hint

   Check the inner product first before proceeding to the Gram-Schmidt process.

6. 6.

   Explain how deriving an orthonormal set can lead to simplifications in a finite element model for structural analysis.

   Hint

   Think about how calculations are interconnected and reliant on the system of vectors.