If $$\vec{a}$$ and $$\vec{b}$$ are perpendicular, then $$\vec{a} \times \left(\vec{a} \times \left(\vec{a} \times (\vec{a} \times \vec{b})\right)\right)$$ is equal to:

Given that $$\vec{a}$$ and $$\vec{b}$$ are perpendicular, we have $$\vec{a} \cdot \vec{b} = 0$$. We need to evaluate $$\vec{a} \times (\vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})))$$.

We use the vector triple product identity $$\vec{a} \times (\vec{a} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{a} - (\vec{a} \cdot \vec{a})\vec{c} = (\vec{a} \cdot \vec{c})\vec{a} - |\vec{a}|^2 \vec{c}$$ repeatedly.

First, $$\vec{a} \times (\vec{a} \times \vec{b}) = (\vec{a} \cdot \vec{b})\vec{a} - |\vec{a}|^2 \vec{b} = 0 - |\vec{a}|^2 \vec{b} = -|\vec{a}|^2 \vec{b}$$.

Next, $$\vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})) = \vec{a} \times (-|\vec{a}|^2 \vec{b}) = -|\vec{a}|^2 (\vec{a} \times \vec{b})$$.

Finally, $$\vec{a} \times (\vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b}))) = \vec{a} \times (-|\vec{a}|^2 (\vec{a} \times \vec{b})) = -|\vec{a}|^2 (\vec{a} \times (\vec{a} \times \vec{b})) = -|\vec{a}|^2 \cdot (-|\vec{a}|^2 \vec{b}) = |\vec{a}|^4 \vec{b}$$.