Equivalence of Definitions of Vector Projection

$(1) \iff (3)$
By definition $1$ of vector projection:


 * $\proj_\mathbf v \mathbf u$ is a like vector to $\mathbf v$ whose length is $\norm {\mathbf u} \cos \theta$

This is obtained by creating a vector quantity:
 * $\paren {\norm {\mathbf u} \cos \theta} \mathbf {\hat v}$

where $\mathbf {\hat v}$ is the unit vector in the direction of $\mathbf v$.

But by :
 * $u_{\parallel \mathbf v} = \norm {\mathbf u} \cos \theta$