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Exercise 3
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Exercise 3
Exercise 3
Exercise 3. (The triangle inequality) Show that dist(p, q) + dist(9,r) > dist(p,r) for all p. q in R". By a metric on a set X we mean a mapping d: X X XR such that 1. d(p, q) > 0, with equality if and only if p=4. 2. d(p.4)=d(9.p). 3. d(p,) + d(9,7) > dp,r). These properties are called, respectively, positive-definiteness, symmetry, and the triangle inequality. The pair (X,d) is called a metric space. Using the above exercise, one immediately checks that (R", dist) is a metric space. Ge- ometry, in its broadest definition, is the study of metric spaces, and Euclidean Geometry, in the modern sense, is the study of the metric space (R", dist). Finally, we define the angle between a pair of nonzero vectors in R" by angle(p, q) : Il ple|| Note that the above is well defined by the Cauchy-Schwartz inequality. Now we have all the necessary tools to prove the most famous result in all of mathematics:
Exercise 3. (The triangle inequality) Show that dist(p, q) + dist(9,r) > dist(p,r) for all p. q in R". By a metric on a set X we mean a mapping d: X X XR such that 1. d(p, q) > 0, with equality if and only if p=4. 2. d(p.4)=d(9.p). 3. d(p,) + d(9,7) > dp,r). These properties are called, respectively, positive-definiteness, symmetry, and the triangle inequality. The pair (X,d) is called a metric space. Using the above exercise, one immediately checks that (R", dist) is a metric space. Ge- ometry, in its broadest definition, is the study of metric spaces, and Euclidean Geometry, in the modern sense, is the study of the metric space (R", dist). Finally, we define the angle between a pair of nonzero vectors in R" by angle(p, q) : Il ple|| Note that the above is well defined by the Cauchy-Schwartz inequality. Now we have all the necessary tools to prove the most famous result in all of mathematics: