## Thomas' Calculus 13th Edition

$(a)\displaystyle \quad (\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}})\cdot{\bf v}$ $(b)\quad {\bf u}\times{\bf v}$ $(c)\quad ({\bf u\times v})\times{\bf v}$ $(d)\quad |({\bf u\times v})\times{\bf w}|$ $(e)\quad ({\bf u\times v})\times({\bf u\times w})$ $(f)\quad |{\bf u} |\displaystyle \cdot\frac{{\bf v}}{|{\bf v}|}$
$(a)$ The projection of ${\bf u}$ onto ${\bf v}$ was given in sec 12-3: $\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}})\cdot{\bf v}$ $(b)$ 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 $|{\bf u} |\displaystyle \cdot\frac{{\bf v}}{|{\bf v}|}$