Orthogonality

Example 4.1.1. (Example 3.5.3 revisited.) Let E = { x = ( x 1 , x 2 , x 3 ) ∈ R 3 $ E = \{ x = (x_{1} , x_{2} , x_{3} ) \in {\mathbb{R}}^{3} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math4_1.tif"/> : x ₁ - 2x ₂ + 3x ₃ = 0} be a subset of R 3 $ {\mathbb{R}}^{3} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math4_2.tif"/> . Then E is a plane in R 3 $ {\mathbb{R}}^{3} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math4_3.tif"/> passing through (0, 0, 0) . For every x = ( x 1 , x 2 , x 3 ) ∈ E $ x = (x_{1} , x_{2} , x_{3} ) \in E $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math4_4.tif"/> we have x ₁ = 2x ₂ - 3x ₃ and x 1 x 2 x 3 = 2 x 2 - 3 x 3 x 2 x 3 = 2 x 2 x 2 0 + 3 x 3 0 x 3 = x 2 2 1 0 + x 3 3 0 1 . $$ \left[ {\begin{array}{*{20}c} {x_{1} } \\ {x_{2} } \\ {x_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {2x_{2} - 3x_{3} } \\ {x_{2} } \\ {x_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {2x_{2} } \\ {x_{2} } \\ 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {3x_{3} } \\ 0 \\ {x_{3} } \\ \end{array} } \right] = x_{2} \left[ {\begin{array}{*{20}c} 2 \\ 1 \\ 0 \\ \end{array} } \right] + x_{3} \left[ {\begin{array}{*{20}c} 3 \\ 0 \\ 1 \\ \end{array} } \right]. $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/math4_1.tif"/>