Question

Дополните векторы до ортонормированного базиса

Answer 1

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Answer 2

$vec{x}=(frac{1}{sqrt3},frac{1}{sqrt3},frac{1}{sqrt3}); ,; ; vec{y}=(frac{1}{sqrt{14}},frac{2}{sqrt{14}},-frac{3}{sqrt{14}})\\|vec{x}|=sqrt{frac{1}{3}+frac{1}{3}+frac{1}{3}}=1; ,; ; |vec{y}|+sqrt{frac{1}{14}+frac{4}{14}+frac{9}{14}}=1\\vec{x}cdot vec{y}=frac{1}{sqrt{42}}+frac{2}{sqrt{42}}-frac{3}{sqrt{42}}=0; ; Rightarrow ; ; vec{x}perp vec{y}\\vec{x}perp vec{z}; ,; ; vec{y}perp vec{z}; ,; ; z=(z_1,z_2,z_3)\\vec{x}cdot vec{z}=frac{z_1}{sqrt3}+frac{z_2}{sqrt3}+frac{z_3}{sqrt3}=0; ; Rightarrow ; ; z_1+z_2+z_3=0$

$vec{y}cdot vec{z}=frac{z_1}{sqrt{14}}+frac{2z_2}{sqrt{14}}-frac{3z_3}{sqrt{14}}=0; ; Rightarrow z_1+2z_2+2z_3=0$

$left { {{z_1+z_2+z_3=0} atop {z_1+2z_2-3z_3=0}} right.; ominus ; left { {{z_1+z_2+z_3=0} atop {z_2-4z_3=0}} right. ; ; left { {{z_1=-z_2-z_3} atop {z_2=4z_3}} right.; left { {{z_1=-5z_3} atop {z_2=4z_3}} right.; Rightarrow \\z_3=t; ; to ; ; pyst; ; t=-1; ,; vec{z}=(5,-4,-1)\\|vec{z}|=sqrt{25+16+1}=42\\vec{z}^circ =(frac{5}{sqrt{42}},-frac{4}{sqrt{42}},-frac{1}{sqrt{42}})\\Otvet:; ; frac{4}{sqrt{42}}.$