Let a unit vector which makes an angle of 60° with $$2\hat{i} + 2\hat{j} - \hat{k}$$ and angle 45° with $$\hat{i} - \hat{k}$$ be $$\overrightarrow{C}$$. Then $$\overrightarrow{C} + \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)$$ is:

We need to find a unit vector $$\overrightarrow{C} = x\hat{i} + y\hat{j} + z\hat{k}$$ such that it makes an angle of 60° with $$\overrightarrow{a} = 2\hat{i} + 2\hat{j} - \hat{k}$$ and an angle of 45° with $$\overrightarrow{b} = \hat{i} - \hat{k}$$.

Since $$\overrightarrow{C}$$ is a unit vector, we have:

$$ x^2 + y^2 + z^2 = 1 \quad \cdots(1) $$

Condition 1: Angle with $$\overrightarrow{a}$$ is 60°.

$$|\overrightarrow{a}| = \sqrt{4 + 4 + 1} = 3$$

$$ \cos 60° = \frac{\overrightarrow{C} \cdot \overrightarrow{a}}{|\overrightarrow{C}||\overrightarrow{a}|} = \frac{2x + 2y - z}{3} $$

$$ \frac{1}{2} = \frac{2x + 2y - z}{3} \implies 2x + 2y - z = \frac{3}{2} \quad \cdots(2) $$

Condition 2: Angle with $$\overrightarrow{b}$$ is 45°.

$$|\overrightarrow{b}| = \sqrt{1 + 1} = \sqrt{2}$$

$$ \cos 45° = \frac{\overrightarrow{C} \cdot \overrightarrow{b}}{|\overrightarrow{C}||\overrightarrow{b}|} = \frac{x - z}{\sqrt{2}} $$

$$ \frac{1}{\sqrt{2}} = \frac{x - z}{\sqrt{2}} \implies x - z = 1 \quad \cdots(3) $$

From (3): $$z = x - 1$$. Substituting into (2):

$$ 2x + 2y - (x - 1) = \frac{3}{2} \implies x + 2y + 1 = \frac{3}{2} \implies x = \frac{1}{2} - 2y \quad \cdots(4) $$

Also, $$z = x - 1 = -\frac{1}{2} - 2y$$. Substituting into (1):

$$ \left(\frac{1}{2} - 2y\right)^2 + y^2 + \left(-\frac{1}{2} - 2y\right)^2 = 1 $$

$$ \frac{1}{4} - 2y + 4y^2 + y^2 + \frac{1}{4} + 2y + 4y^2 = 1 $$

$$ 9y^2 + \frac{1}{2} = 1 \implies y^2 = \frac{1}{18} \implies y = \pm \frac{1}{3\sqrt{2}} $$

Case 1: $$y = \frac{1}{3\sqrt{2}}$$, then $$x = \frac{1}{2} - \frac{\sqrt{2}}{3}$$, $$z = -\frac{1}{2} - \frac{\sqrt{2}}{3}$$.

Case 2: $$y = -\frac{1}{3\sqrt{2}}$$, then $$x = \frac{1}{2} + \frac{\sqrt{2}}{3}$$, $$z = -\frac{1}{2} + \frac{\sqrt{2}}{3}$$.

Now compute $$\overrightarrow{C} + \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)$$ for each case.

Case 2 check:

$$i$$-component: $$\frac{1}{2} + \frac{\sqrt{2}}{3} - \frac{1}{2} = \frac{\sqrt{2}}{3}$$

$$j$$-component: $$-\frac{1}{3\sqrt{2}} + \frac{1}{3\sqrt{2}} = 0$$

$$k$$-component: $$-\frac{1}{2} + \frac{\sqrt{2}}{3} - \frac{\sqrt{2}}{3} = -\frac{1}{2}$$

This gives $$\frac{\sqrt{2}}{3}\hat{i} - \frac{1}{2}\hat{k}$$, which matches Option 1.

Hence the correct answer is Option 1.