Putting these together, we compute that \begin{align}c \bfx &= \langle c x_1, c x_2, c x_3 \rangle\\ \left | c \bfx \right | &= \sqrt{(c x_1)^2 + (c x_2)^2 + (c x_3)^2} \end{align}
In order to relate $\left | c \bfx \right|$ to $\left | \bfx \right|$ we try to factor out $c$:
\begin{align}\left | c \bfx \right | &= \sqrt{c^2 (x_1^2 + x_2^2 + x_3)}\\ &= \sqrt{c_\phantom{1 }^2} \sqrt{x_1^2 + x_2^2 + x_3^2} \\ &= \left | c \right | \left | \bfx \right |\end{align} where $\left| c \right|$ is the absolute value of $c$.
