Vectors $$a\hat{i} + b\hat{j} + \hat{k}$$ and $$2\hat{i} - 3\hat{j} + 4\hat{k}$$ are perpendicular to each other when $$3a + 2b = 7$$, the ratio of $$a$$ to $$b$$ is $$\frac{x}{2}$$. The value of $$x$$ is _____.

The vectors $$a\hat{i} + b\hat{j} + \hat{k}$$ and $$2\hat{i} - 3\hat{j} + 4\hat{k}$$ are perpendicular, so their dot product is zero: $$2a - 3b + 4 = 0 \quad \cdots (1)$$.

Using the given condition $$3a + 2b = 7 \quad \cdots (2)$$, we solve the system of equations. From (1) we have $$2a - 3b = -4$$. Multiplying (2) by 3 gives $$9a + 6b = 21$$ and multiplying (1) by 2 gives $$4a - 6b = -8$$. Adding these equations yields $$13a = 13 \implies a = 1$$, and substituting back into (2) leads to $$3(1) + 2b = 7 \implies b = 2$$.

Finally, the ratio $$\frac{a}{b} = \frac{1}{2} = \frac{x}{2}$$ implies $$x = 1$$. The answer is $$\boxed{1}$$.