Proof of $\langle v, u \rangle u = (uu^T)v$

We were given this identity during the lecture, but the proof was omitted:

$$\langle v, u \rangle u = (uu^T)v \text{ with } ||u||=1$$ If I write out the intermediate matrices, I see that the equality holds, but I wanted to prove it more analytically.

I tried starting with $$\langle v, u \rangle u = (v^Tu)u$$ using the definition of dot product, then the professor suggested we use associativity of matrix/vector multiplication to rewrite the operations as an outer product.

I'm not very familiar with the outer product and I do not get how to do use it for the last part. Any insight appreciated.

The inner product is scalar valued, so can be placed on the right of $$u$$ instead of on the left. We can also write it as $$u^Tv$$ instead, because the product is symmetric. The associativity of matrix multiplication completes the proof.