Solution
- The question would like us to find the projection of u on v given that
[tex]\begin{gathered} u=\langle12,-5\rangle \\ v=\langle3,9\rangle \end{gathered}[/tex]
Explanation
- The projection of u on v is defined as follows
[tex]\frac{\vec{u}\text{.}\vec{v}}{||v\mleft\Vert\mright?^2}\times\vec{v}[/tex]
- Thus, we can simply evaluate the projection of u on v using the above definition as follows:
[tex]\begin{gathered} \vec{u}=\langle-12,-5\rangle\text{ can be rewritten as follows:} \\ \vec{u}=-12i-5j \\ \\ \vec{v}=\langle3,9\rangle\text{ can be rewritten as follows:} \\ \vec{v}=3i+9j \\ \\ \text{Thus,} \\ \vec{u}\text{.}\vec{v}=(-12i-5j)(3i+9j) \\ \vec{u}\text{.}\vec{v}=-36i^2-45j^2=-36-45=-81\text{ (Since }i^2=j^2=1) \\ \\ \begin{matrix} \\ \mleft\Vert\vec{v}\mright?\mleft\Vert \mright?^2=(3i+9j)^2=(3i+9j)(3i+9j)=9+81=90\end{matrix} \\ \\ \therefore\text{The projection of u on v is:} \\ \frac{\vec{u}\text{.}\vec{v}}{||v\mleft\Vert\mright?^2}\times\vec{v}=-\frac{81}{90}\langle3,9\rangle=-\frac{9}{10}\langle3,9\rangle=\langle-\frac{27}{10},-\frac{81}{10}\rangle \end{gathered}[/tex]
Final Answer
The answer is
[tex]\langle-\frac{27}{10},-\frac{81}{10}\rangle[/tex]
