27.6. Triangle Inequality

Try these first

1. 1.

   State the Triangle Inequality for vectors u and v.

   Hint

   Think about the vector norms.

2. 2.

   If u = (3, 4) and v = (6, 8), what are ||u|| and ||v||?

   Hint

   Use the Pythagorean theorem to find these lengths.

3. 3.

   What does the Triangle Inequality state?

   • ||u - v|| < ||u|| + ||v||
   • ||u + v|| ≥ ||u|| + ||v||
   • ||u + v|| ≤ ||u|| + ||v||

   Hint

   Think about the lengths of vectors.

4. 4.

   The Triangle Inequality holds true for all inner product spaces.

   • True
   • False

   Hint

   Is this a general rule or an exception?

5. 5.

   Given vectors u = (3, 1) and v = (-1, 4), prove the Triangle Inequality using their norms.

   Hint

   First find the result of u + v, then calculate the norms.

6. 6.

   Prove or disprove: If ||u|| > 0 and ||v|| > 0, then ||u + v|| > ||u|| + ||v||?

   Hint

   Consider the definition and properties of norms.