Orthogonal Projection
of vector $\mathbf{b}$ to vector $\mathbf{a}$
_{Mnemonic: mapping over magnitude} $$\beta=\frac{\mathbf{a}^T\mathbf{b}}{\mathbf{a}^T\mathbf{a}}$$

Orthogonal Vector Decomposition
of vector $\mathbf{t}$ by reference vector $\mathbf{r}$
$$\mathbf{t}_{\|}=\mathbf{r}\frac{\mathbf{t}^T\mathbf{r}}{\mathbf{r}^T\mathbf{r}}$$ $$\mathbf{t}_{\perp}=\mathbf{t}-\mathbf{t}_{\|}$$