= Solution
Take the <dot product> of the equation with $a$. Since $a\cdot(a\times r)=0$, we obtain $a\cdot r=(a\cdot c)/\lambda$. Next take its <cross product> with $a$ and apply the <vector triple product identity>:
$$
a(a\cdot r)-|a|^2r+\lambda(a\times r)=a\times c.
$$
Substitute $a\times r=c-\lambda r$ and the known <dot product>, then rearrange:
$$
\boxed{r=\frac{\lambda c-a\times c+\lambda^{-1}(a\cdot c)a}{\lambda^2+|a|^2}.}
$$
Here the <vectors> and $\lambda\ne0$ are real, so the denominator is positive. The formula also covers $a=0$, giving $r=c/\lambda$. To check existence as well as uniqueness, write its numerator as $N$. The <vector triple product identity> gives $a\times N=\lambda a\times c-a(a\cdot c)+|a|^2c$, so $a\times N+\lambda N=(\lambda^2+|a|^2)c$. Substitution verifies the original equation. This is an instance of the <inverse of a scalar plus a cross-product operator>.
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