How do you project vectors?

Question

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!

Answer 1

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 $$ proj_\mathbf{w} \mathbf{v} = \left| {proj_\mathbf{w} \mathbf{v}} \right|\frac{\mathbf{w}} {{\left| \mathbf{w} \right|}} = \left( {\frac{{\mathbf{v} \cdot \mathbf{w}}} {{\left| \mathbf{w} \right|^2 }}} \right)\mathbf{w} = \left( {\frac{{\mathbf{v} \cdot \mathbf{w}}} {{\mathbf{w} \cdot \mathbf{w}}}} \right)\mathbf{w} $$

Answer 2

Let $\mathbf w'$ be a vector orthogonal to $\mathbf w$, for example $\mathbf w'=(2,-1)$.

Find coefficients $\alpha$ and $\beta$ such that $\mathbf v=\alpha\mathbf w+\beta\mathbf w'$. The vector $\alpha\mathbf w$ is the projection.

The second question looks the same, but note that $\mathbf v$ and $\mathbf w$ are linealry dependent.