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 _____.

For perpendicular vectors the dot product is zero:
$$ (a\hat{i}+b\hat{j}+\hat{k}) \cdot (2\hat{i}-3\hat{j}+4\hat{k}) = 2a-3b+4 = 0 $$

We have $$3a+2b=7$$ and from the dot product $$2a=3b-4 \Rightarrow a=(3b-4)/2$$.

Substituting into this gives $$3(3b-4)/2+2b=7 \Rightarrow 13b=26 \Rightarrow b=2, a=1$$.

Since $$a/b = 1/2 = x/2$$, it follows that $$x = 1$$.