Question Number 10562 by FilupS last updated on 18/Feb/17
![x= [(x_1 ),(x_2 ),(( ⋮)),(x_n ) ] y= [(y_1 ),(y_2 ),(( ⋮)),(y_n ) ] x, y ∈ R^n 1. Prove the length of the vector x, denoted ∣∣x∣∣, is equal to (√(x_1 ^2 +x_2 ^2 +...+x_n ^2 )) 2. Determine if ∣∣x−y∣∣=∣∣y−x∣∣](Q10562.png)
$$\boldsymbol{{x}}=\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{\:\vdots}\\{{x}_{{n}} }\end{bmatrix}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{y}}=\begin{bmatrix}{{y}_{\mathrm{1}} }\\{{y}_{\mathrm{2}} }\\{\:\vdots}\\{{y}_{{n}} }\end{bmatrix}\:\:\:\:\:\:\:\:\boldsymbol{{x}},\:\boldsymbol{{y}}\:\in\:\mathbb{R}^{{n}} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vector}\:\boldsymbol{{x}},\:\mathrm{denoted}\:\mid\mid\boldsymbol{{x}}\mid\mid, \\ $$$$\:\:\:\:\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\sqrt{{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} } \\ $$$$\mathrm{2}.\:\mathrm{Determine}\:\mathrm{if}\:\:\:\mid\mid\boldsymbol{{x}}−\boldsymbol{{y}}\mid\mid=\mid\mid\boldsymbol{{y}}−\boldsymbol{{x}}\mid\mid \\ $$