23.15. Orthogonality and Linear Independence

Try these first

1. 1.

   Define orthogonality in your own words.

   Hint

   Think about the geometric interpretation of two lines.

2. 2.

   Are the vectors [1, 2] and [-2, 1] orthogonal? Justify your answer.

   Hint

   Calculate the dot product to check.

3. 3.

   What does it mean for two vectors to be orthogonal?

   • They have the same direction
   • Their dot product is zero
   • They are linearly dependent

   Hint

   Think about the geometric relationship between the vectors.

4. 4.

   Is it true or false? Orthogonal vectors cannot be linearly dependent.

   • True
   • False

   Hint

   Consider the properties of independence!

5. 5.

   Given the vectors A = [3, -4], B = [4, 2], and C = [0, 0], determine if they are linearly independent. Provide a justification.

   Hint

   Calculate dot products and assess the coefficients.

6. 6.

   In a mechanical system, explain how having linearly dependent vectors can impact the solution to structural analysis problems.

   Hint

   Consider the equations in equilibrium and how they relate to the forces in the structure.