21.9.3. Gram-Schmidt Process

Try these first

1. 1.

   Define orthogonal vectors.

   Hint

   Think about the angle between them.

2. 2.

   What does it mean for a set of vectors to be orthonormal?

   Hint

   One condition is that they shouldn't influence each other, and another is that their length should be one.

3. 3.

   What does it mean for two vectors to be orthogonal?

   • They are the same
   • Their dot product is zero
   • They are parallel

   Hint

   Think about how you can prove two vectors don't affect each other directionally.

4. 4.

   True or False? The Gram-Schmidt process can only be applied to vectors in R^3.

   • True
   • False

   Hint

   Consider the definition of the process and its universality.

5. 5.

   Given the vectors v1 = (3, 1) and v2 = (2, 2), apply the Gram-Schmidt Process to create an orthonormal set.

   Hint

   Remember the steps of normalization and projection for each vector.

6. 6.

   Explain the importance of using orthonormal vectors in stability analyses in civil engineering.

   Hint

   Think about how simpler calculations lead to more reliable models in engineering contexts.