Question: This is the first time I have attempted such a problem, and have been tackling it all morning. I've been trying to use simultaneous equations to solve for the variables, but it doesn't seem to be working out, so perhaps that's not the way to go...

$$\hat{\bf a} = \dfrac{1}{\sqrt{14}}({\bf i}+2{\bf j}+3{\bf k})$$
$$\hat{\bf b} = \dfrac{1}{\sqrt{5}}(2{\bf i}-{\bf j})$$
$$\hat{\bf c} = \dfrac{1}{\sqrt{70}}(3{\bf i}+6{\bf j}-5{\bf k})$$

Express the vector ${\bf d} = 5{\bf i}+3{\bf j}+2{\bf k}$ in the new $(\hat{\bf a},\hat{\bf b},\hat{\bf c})$ basis.

My attempted solution can be seen here: <https://drive.google.com/file/d/0B8CnEriPXOd1T3Z1amU4UGlMVU0/view>

There is basic arithmetic mistake in my attempted answer, so I have answered my own question below. Sorry for wasting anybody's time; I'll try to be more careful.

Answer: 
Wow, after all of that, and what I had believed to be a meticulous examination of my work, I rechecked it in the morning after some refreshing sleep, and found a basic arithmetic mistake. Theta should have been (17 x rt(14))/14. Sorry to have wasted anybody's time!
You're looking for $[r,s,t]$ where $r\hat{\bf a}+s\hat{\bf b}+t\hat{\bf c}=5{\bf i}+3{\bf j}+2{\bf k}$.

This means that $\dfrac{r}{\sqrt{14}}+\dfrac{2s}{\sqrt{5}}+\dfrac{3t}{\sqrt{70}}=5$ (the coefficients of ${\bf i}$). Same for the coefficients of ${\bf j}$ and ${\bf k}$. Then it's just a matter of slugging through this rather unattractive system of equations. :(