Exercise 21.  If you study carefully the proof of the triangle inequality, you will note that the reasons for the inequality hinge on  .  Under what conditions will these two quantities be equal, thus turning the triangle inequality into an equality ?

Solution 21.

See text and/or instructor's solution manual.

Let    and  .  We have  .  

Then    and  ,  and we have  

          

If either or equals 0, then clearly  .  

If neither equals 0, then    precisely when  .  

This occurs when the points and lie on a straight line through the origin.  Show the details for this last statement.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell