Let $$\vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k}$$, $$\vec{b} = \hat{i} + \hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 3\hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$|\vec{r}|$$ is equal to:

Given $$\vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k}$$, $$\vec{b} = \hat{i} + \hat{k}$$, and $$\vec{c} = \hat{i} + 2\hat{j} - 3\hat{k}$$. We need to find $$|\vec{r}|$$ where $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$.

From the condition $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ we have $$\vec{r} \times \vec{a} - \vec{c} \times \vec{a} = \vec{0}$$, which implies $$(\vec{r} - \vec{c}) \times \vec{a} = \vec{0}$$. Hence $$\vec{r} - \vec{c}$$ is parallel to $$\vec{a}$$, and we can write $$\vec{r} = \vec{c} + \lambda \vec{a}$$ for some scalar $$\lambda$$.

Substituting the vectors gives $$\vec{r} = (1 + 2\lambda)\hat{i} + (2 - 7\lambda)\hat{j} + (-3 + 5\lambda)\hat{k}$$.

Applying the dot product condition $$\vec{r} \cdot \vec{b} = 0$$ yields $$(1 + 2\lambda)\cdot 1 + (2 - 7\lambda)\cdot 0 + (-3 + 5\lambda)\cdot 1 = 0$$, so $$(1 + 2\lambda) + (-3 + 5\lambda) = 0$$ which leads to $$-2 + 7\lambda = 0$$ and hence $$\lambda = \frac{2}{7}$$.

Substituting $$\lambda = \frac{2}{7}$$ gives $$\vec{r} = \left(1 + \frac{4}{7}\right)\hat{i} + (2 - 2)\hat{j} + \left(-3 + \frac{10}{7}\right)\hat{k} = \frac{11}{7}\hat{i} + 0\hat{j} - \frac{11}{7}\hat{k}$$.

The magnitude of $$\vec{r}$$ is $$|\vec{r}| = \sqrt{\left(\frac{11}{7}\right)^2 + 0 + \left(\frac{11}{7}\right)^2} = \frac{11}{7}\sqrt{2}$$, so $$|\vec{r}| = \frac{11\sqrt{2}}{7} \approx 2.22$$. Therefore, $$|\vec{r}| = \frac{11\sqrt{2}}{7}$$.