21.9. Orthogonality and Gram-Schmidt Process

Try these first

1. 1.

   Define orthogonal vectors.

   Hint

   Think about the geometric interpretation and the dot product.

2. 2.

   What is the first step in the Gram-Schmidt process?

   Hint

   Consider what normalization means for a vector.

3. 3.

   What defines two vectors as orthogonal?

   • They are parallel
   • Their lengths are equal
   • Their dot product is zero

   Hint

   Focus on the geometric interpretation.

4. 4.

   True or False: Orthonormal vectors can have lengths greater than one.

   • True
   • False

   Hint

   Recall the definition of unit vectors.

5. 5.

   Given the set of vectors A = {(1, 2, 0), (2, 1, 0)}, apply the Gram-Schmidt process and find an orthonormal basis.

   Hint

   Start by calculating the lengths and projections carefully.

6. 6.

   Explain how the concepts of orthogonality and the Gram-Schmidt process enhance numerical solutions in engineering fields.

   Hint

   Consider examples of calculations in structural analysis.