The definition u v = |u| |v| cos implies that |u v| |u||v|
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The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences.
Consider the vectors u, v, and u + v (in any number of dimensions). Use the following steps to prove that |u + v| ≤ |u | + |v|.
a. Show that |u + v|2 = (u + v) • (u + v) = |u|2 + 2u • v + |v|2.
b. Use the Cauchy–Schwarz Inequality to show that |u + v|2 ≤ (| u| + |v|)2.
c. Conclude that |u + v| ≤ |u| + |v|.
d. Interpret the Triangle Inequality geometrically in R2 or R3.
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Related Book For
Calculus Early Transcendentals
ISBN: 978-0321947345
2nd edition
Authors: William L. Briggs, Lyle Cochran, Bernard Gillett
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