24.17. Orthogonality in Vector Spaces

Try these first

1. 1.

   Define orthogonal vectors.

   Hint

   Think about their geometric representation.

2. 2.

   What condition must be met for vectors to be orthonormal?

   Hint

   Recall the definition of unit length.

3. 3.

   What does it mean for two vectors to be orthogonal?

   • Their dot product is one.
   • Their dot product is zero.
   • They have the same direction.

   Hint

   Think about how two vectors' directions relate in space.

4. 4.

   True or False: All orthonormal vectors are orthogonal.

   • True
   • False

   Hint

   Recall the definition of orthonormal sets.

5. 5.

   Given vectors A = (2, 3, 6) and B = (1, -1, 2), are they orthogonal? Verify your answer.

   Hint

   Remember to calculate the dot product correctly.

6. 6.

   Find a new orthonormal basis for the vectors in R² given by (4, 3) and (1, 1) using the Gram-Schmidt process.

   Hint

   Follow the Gram-Schmidt steps methodically.