27.4. Orthogonality and Orthonormality

Try these first

1. 1.

   Define orthogonal vectors.

   Hint

   Think about their angle with respect to each other.

2. 2.

   What conditions must a set of vectors meet to be considered orthonormal?

   Hint

   Recall the properties of orthogonality and length.

3. 3.

   What defines two vectors as orthogonal?

   • Their lengths are equal
   • Their inner product equals zero
   • They lie in the same direction

   Hint

   Think about what it means for two lines to meet.

4. 4.

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

   • True
   • False

   Hint

   Consider the criteria for orthonormal sets.

5. 5.

   Given two vectors u = (2, -1) and v = (-0.5, 1) in R^2, determine if they are orthogonal.

   Hint

   Find the inner product to check their orthogonality.

6. 6.

   Assert whether the following set of vectors forms an orthonormal set: A = { (1,0), (0,1), (1,1)/√2, (1,-1)/√2 }.

   Hint

   Evaluate each vector's length and their pairwise inner products.