In every triangle. the length of any side is less than the

Chapter , Problem 125

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In every triangle. the length of any side is less than the sum of the lengths of the other two sides. When this observation is gcncraliu:d ton. we obtain the triallgl~ inequality (Theorem 6.4 ). "hich "Jte' [ u + v ~ l ull+ II,. for any vectors u nnd ,. in n. Let 4.01 ~ [il -[=~] [ 2.01 ] v, = 6.01 . [ 3.0 1] 6.0 1 " 2 = 9.0 1 . 12.0 1 8.01 (a) Verify the triangle inequality for u and '' (b) Verify the tri:mglc inc4uality for u and v1. (c) Verify the triangle inequality for u tmd v2. and (d) From what you have observed in (b) and (c). make a conjecture about when ec1uality occurs in the triangle inequality. (e) lmerpret your conjecture in {tl) geometrically in n 2