Answer
$(a)\displaystyle \quad (\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}})\cdot{\bf v}$
$(b)\quad {\bf u}\times{\bf v}$
$(c)\quad ({\bf u\times v})\times{\bf v}$
$(d)\quad |({\bf u\times v})\times{\bf w}|$
$(e)\quad ({\bf u\times v})\times({\bf u\times w})$
$(f)\quad |{\bf u} |\displaystyle \cdot\frac{{\bf v}}{|{\bf v}|}$
Work Step by Step
$(a)$
The projection of ${\bf u}$ onto ${\bf v}$ was given in sec 11-3:
$\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}})\cdot{\bf v}$
$(b)$
The cross product of two vectors is a vector orthogonal to both.
${\bf u}\times{\bf v}$
$(c)$
The cross product of two vectors is a vector orthogonal to both.
$({\bf u\times v})\times{\bf v}$
$(d)$
The volume of the parallelepiped equals the absolute value of the triple scalar product.
$|({\bf u\times v})\times{\bf w}|$
$(e)$
The cross product of two vectors is a vector orthogonal to both.
$({\bf u\times v})\times({\bf u\times w})$
$(f)$
The direction of ${\bf v}$ is the unit vector $\displaystyle \frac{{\bf v}}{|{\bf v}|}$,
and if the length is to be $|{\bf u} |$, the vector is
$|{\bf u} |\displaystyle \cdot\frac{{\bf v}}{|{\bf v}|}$
