# Map preserves angle iff scalar multiple of isometry

How do I prove that a map preserves the angle if and only if it's the scalar multiple of an isometry.

I get the "if" direction by using definition of isometry.

How do I show the other direction, i.e. that $$\frac{}{\left\lVert u\right\rVert\left\lVert v\right\rVert} = \frac{}{\left\lVert L(u)\right\rVert\left\lVert L(v)\right\rVert}$$ implies scalar multiple of isometry?

• Hint: let $v=u$. – S. Dewar Jan 17 at 21:53