User:Tetrahedron93/Sandbox

Let there be a vector $$v= $$, defined with a metric $$|v| = \sqrt{\sum a_ix_i^2}$$.

$$|\lambda v|= \sqrt{\sum a_i {(\lambda x_i)}^2}$$

$$= \sqrt{\sum a_i \lambda^2 x_i^2}$$

$$= \sqrt{\lambda^2 \sum a_i x_i^2}$$

$$= \lambda \sqrt{\sum a_i x_i^2}$$

$$= \lambda |v|$$

And so scaling the length of the vector requires only component-wise scaling.