4d4983…: “Left of the fair sphere”

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Suppose that $\vec {u}$ and $\vec {v}$ are nonzero three dimensional vectors and let $\dotp$ denote the vector dot product and $\cdot$ denote scalar multiplication. Select all of the following that must be true. Try to think about each statement both algebraically and geometrically.

$\scal _\vec {u}(\vec {u}) = \vec {u}$ . If $\vec {u}$ and $\vec {v}$ are parallel, then $\proj _\vec {v}(\vec {u}) = \vec {u}$ . If $\vec {v}$ and $\vec {w}$ are parallel, then $\proj _\vec {v}(\vec {u}) = \proj _\vec {w}(\vec {u})$ . If $\proj _\vec {v}(\vec {u}) \dotp \vec {v} = \vec {0}$ , then $\vec {u}$ and $\vec {v}$ are parallel. $\vec {u}+\left |\proj _\vec {v}(\vec {u})\right |$ is defined. If $\uvec {w}$ is a unit vector, then $\vec {u} \dotp \uvec {w} = \scal _\uvec {w}(\vec {u})$ .