The vectors are given to be:
[tex]\begin{gathered} u=\langle-10,-3\rangle \\ v=\langle4,8\rangle \end{gathered}[/tex]The question asks to evaluate the following projection:
[tex]proj_vu[/tex]To do this, we will use the formula:
[tex]proj_vu=(\frac{u\cdot v}{||v||^2})v[/tex]We can evaluate the following:
[tex]\begin{gathered} u\cdot v=\langle-10,-3\rangle\cdot\langle4,8\rangle=(-10)\cdot(4)+(-3)\cdot(8) \\ u\cdot v=-64 \end{gathered}[/tex]and
[tex]\begin{gathered} ||v||=\sqrt{|4|^2+|8|^2}=\sqrt{80} \\ ||v||=4\sqrt{5} \\ \therefore \\ ||v||^2=(4\sqrt{5})^2=80 \end{gathered}[/tex]Therefore, the projection is calculated to be:
[tex]\begin{gathered} proj_vu=\frac{-64}{80}\cdot\langle4,8\rangle \\ proj_vu=-\frac{4}{5}\cdot\langle4,8\rangle \\ proj_vu=\langle-\frac{16}{5},-\frac{32}{5}\rangle \end{gathered}[/tex]The LAST OPTION is correct.
