Question

Given vector u equals open angled bracket negative 10 comma negative 3 close angled bracket and vector v equals open angled bracket 4 comma 8 close angled bracket comma what is projvu?open angled bracket negative 640 over 109 comma negative 192 over 169 close angled bracketopen angled bracket negative 256 over 109 comma negative 512 over 109 close angled bracketopen angled bracket negative 8 comma negative 3 close angled bracketopen angled bracket negative 16 over 5 comma negative 32 over 5 close angled bracket

Answer 1

The vectors are given to be:

`\begin{gathered} u=\langle-10,-3\rangle \\ v=\langle4,8\rangle \end{gathered}`

The question asks to evaluate the following projection:

`proj_vu`

To do this, we will use the formula:

`proj_vu=((u\cdot v)/(||v||^2))v`

We can evaluate the following:

`\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}`

and

`\begin{gathered} ||v||=√(|4|^2+|8|^2)=√(80) \\ ||v||=4√(5) \\ \therefore \\ ||v||^2=(4√(5))^2=80 \end{gathered}`

Therefore, the projection is calculated to be:

`\begin{gathered} proj_vu=(-64)/(80)\cdot\langle4,8\rangle \\ proj_vu=-(4)/(5)\cdot\langle4,8\rangle \\ proj_vu=\langle-(16)/(5),-(32)/(5)\rangle \end{gathered}`

The LAST OPTION is correct.