writing a vector in a specific form based on restrictions

Question

Given $\vec{a}=[2,1,1]$ and $\vec{b}=[3,2,5]$. Write $\vec{a}$ in the form $\vec{a}=\vec{a_1}+\vec{a_2}$ where $\vec{a_1}$ is parallel to $\vec{b}$ and $\vec{a_2}$ is orthogonal to $\vec{b}$

I'm not quite sure what to do this question. I'm pretty sure that $\vec{a_1}\times\vec{b}=0$ and $\vec{a_2}\cdot\vec{b}=0$ Furthermore, if we let $\vec{a_1}=[x_1,y_1,z_1]$ and $\vec{a_2}=[x_2,y_2,z_2]$ then $x_1+x_2=2, y_1+y_2=1, z_1+z_2=1$. However, even with this information it still feels like I'm just guessing and checking different values and seeing if they work.

Answer 1

HINT:

With $\vec{a_1}||\vec{b}, \vec{a_2}\perp \vec{b}$, set $\vec{a_1}=[3x,2x,5x], \vec{a_2}=[a,b,c] \to $ \begin{align}\vec{a}=\vec{a_1}+\vec{a_2}=[3x+a,2x+b,5x+c]=[2,1,1]\end{align}\begin{align}3x+a=2 \tag 1,2x+b=1,5x+c=1\end{align} \begin{align}\vec{a_2}\cdot\vec{b}=[a,b,c]\cdot[3,2,5]=3a+2b+5c=0 \tag 2\end{align}

Given 4 linear equations (1) and (2) with 4 unknown variables, you can solve these equations from here.

Answer 2

Write $\vec a$ as$$\left(\vec a.\frac{\vec b}{\left\|\vec b\right\|}\right)\frac{\vec b}{\left\|\vec b\right\|}+\left(\vec a-\left(\vec a.\frac{\vec b}{\left\|\vec b\right\|}\right)\frac{\vec b}{\left\|\vec b\right\|}\right)=\frac1{38}(39,26,65)+\frac1{38}(37,12,-27).$$Can you check that the first of these vectors is parallel to $\vec b$ and that the second one is orthogonal to it?