21.9.2. Orthonormal Set

Try these first

1. 1.

   Define an orthonormal set.

   Hint

   Think about the definitions of both orthogonal and unit vectors.

2. 2.

   What is the dot product of orthogonal vectors?

   Hint

   Orthogonal means at right angles; what happens when you multiply their components?

3. 3.

   What characterizes an orthonormal set?

   • Vectors are parallel
   • Vectors are orthogonal and unit vectors
   • Vectors have arbitrary lengths

   Hint

   Think of the definitions of orthogonal and unit.

4. 4.

   True or False: The Gram-Schmidt process can be used to convert a set of dependent vectors into an orthonormal set.

   • True
   • False

   Hint

   Recall the conditions necessary for applying the Gram-Schmidt process.

5. 5.

   Using the vectors v1 = (3, 1) and v2 = (2, 2), apply the Gram-Schmidt process and derive the orthonormal set.

   Hint

   Start by calculating the lengths and projections step by step.

6. 6.

   Discuss a scenario in engineering where using an orthonormal set would help in simplifying complex calculations.

   Hint

   Think about how simplifying computations streamlines structural modeling.