We have
| v | = 1 2 + 2 2 + 3 2 + 4 2 = 30 {\displaystyle |{\mathbf {v}}|={\sqrt {1^{2}+2^{2}+3^{2}+4^{2}}}={\sqrt {30}}}
and
| x | = a 2 + a 2 + a 2 + a 2 = 4 a 2 = 4 a 2 = 2 | a | {\displaystyle |{\mathbf {x}}|={\sqrt {a^{2}+a^{2}+a^{2}+a^{2}}}={\sqrt {4a^{2}}}={\sqrt {4}}{\sqrt {a^{2}}}=2|a|}
So | v | = | x | {\displaystyle |{\mathbf {v}}|=|{\mathbf {x}}|} exactly when 30 = 2 | a | {\displaystyle {\sqrt {30}}=2|a|} and so
a = ± 30 2 {\displaystyle \color {blue}a=\pm {\frac {\sqrt {30}}{2}}}
exactly when 
and so
