<s>[INST] How do you project vectors? So, I am relatively new to the concept of vectors. I'm not to sure on how to "project a vector", as shown below.
For $\textbf{v} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}$ and $\textbf{w} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, compute $\text{proj}_{\textbf{w}} \textbf{v}$.
And... For $\textbf{v} = \begin{pmatrix} -10 \\ 6 \end{pmatrix}$ and $\textbf{w} = \begin{pmatrix} 15 \\ -9 \end{pmatrix}$, compute $\text{proj}_{\textbf{w}} \textbf{v}$.
These questions are virtually the same, but I don't know how to calculate the projections, thanks!
 [/INST] Recall what is the application of the dot product between vectors in physics:
$$
\mathbf{v} \cdot \mathbf{w} = \text{scalar} = \left| {proj_\mathbf{w} \mathbf{v}} \right| \cdot \left| \mathbf{w} \right|
$$
then
$$
\left| {proj_\mathbf{w} \mathbf{v}} \right| = \frac{{\mathbf{v} \cdot \mathbf{w}}}
{{\left| \mathbf{w} \right|}}
$$
and
$$
proj_\mathbf{w} \mathbf{v} = \left| {proj_\mathbf{w} \mathbf{v}} \right|\frac{\mathbf{w}}
{{\left| \mathbf{w} \right|}} = \left( {\frac{{\mathbf{v} \cdot \mathbf{w}}}
{{\left| \mathbf{w} \right|^2 }}} \right)\mathbf{w} = \left( {\frac{{\mathbf{v} \cdot \mathbf{w}}}
{{\mathbf{w} \cdot \mathbf{w}}}} \right)\mathbf{w}
$$
</s>