MathDB
vector inequality involving normed and inner products

Source: SEEMOUS 2011 P3

June 18, 2021
vectorinequalitieslinear algebra

Problem Statement

Given vectors a‾,b‾,c‾∈Rn\overline a,\overline b,\overline c\in\mathbb R^n, show that (∥a‾∥⟨b‾,c‾⟩)2+(∥b‾∥⟨a‾,c‾⟩)2≤∥a‾∥∥b‾∥(∥a‾∥∥b‾∥+∣⟨a‾,b‾⟩∣)∥c‾∥2(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2where ⟨x‾,y‾⟩\langle\overline x,\overline y\rangle denotes the scalar (inner) product of the vectors x‾\overline x and y‾\overline y and ∥x‾∥2=⟨x‾,x‾⟩\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle.
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