In the long paragraph above equation $(2)$, http://mathinsight.org/dot_product avers:
This leads to the definition that the dot product $\mathbf{a⋅b}$, divided by $∥\mathbf{b}∥$ (= magntitude of $\mathbb{b}$), is the projection of $\mathbf{a}$ onto $\mathbf{b}$: $\text{projection of $\mathbf{a}$ onto } \mathbf{b} = \color{green}{\dfrac{\mathbf{a}\cdot \mathbf{b}}{\|\mathbf{b}\|}} = \|\mathbf{a}\| \cos\theta.$
Yet P27 in David Poole avouches:
If $\mathbf{a \neq 0,b} \in \mathbb{R^n}$, then $\text{projection of $\mathbf{a}$ onto } \mathbf{b} = proj_b(\mathbf{a}) = \dfrac{\mathbf{b}\cdot \mathbf{a}}{\mathbf{b} \cdot \mathbf{b}}\mathbf{b} = \color{green}{\dfrac{\mathbf{b}\cdot \mathbf{a}}{|\mathbf{b}|}}\dfrac{\mathbf{b}}{|\mathbf{b}|} .$
Would someone please reconcile these two definitions? Why is Poole's normalised (thanks to $\dfrac{\mathbf{b}}{|\mathbf{b}|}$)? mathinsight.org contains pictures so I thought to include Poole's:
