Solutions

Abstract

(a) (A.4), (A.3):

$$\varvec{ 0 } = \varvec{ 0 } + \left( -\varvec{ 0 }\right) = -\varvec{ 0 }. $$

(b) (A.1)–(A.4), (B.3):

$$\begin{aligned} \alpha \varvec{ 0 } = \varvec{ 0 } + \alpha \varvec{ 0 }= & {} \alpha \varvec{x} + \left( - \alpha \varvec{x}\right) + \alpha \varvec{ 0 } \nonumber \\= & {} \alpha \left( \varvec{ 0 } + \varvec{x}\right) + \left( - \alpha \varvec{x}\right) = \alpha \varvec{x} + \left( - \alpha \varvec{x}\right) = \varvec{ 0 }. \nonumber \end{aligned}$$