No, you can't cancel the dot product.

It's sometimes hard to keep track of which variables are "known" and which are "unknown". Here you are trying to express some vector x (which is therefore "known") as a combination of $\displaystyle a \times b$, $\displaystyle a \times c$, and $\displaystyle b \times c$, where a, b, and c are "known". The other variables - $\displaystyle r_1$, $\displaystyle r_2$, and $\displaystyle r_3$ are the coefficients we're looking for - they are "unknown".

So all you have to do is divide to get your answer:

$\displaystyle r_3 = \frac{x \cdot a}{a \cdot (b \times c)}$

$\displaystyle r_2 = \frac{x \cdot b}{b \cdot (a \times c)}$

$\displaystyle r_1 = \frac{x \cdot c}{c \cdot (a \times b)}$

-Hollywood