Let's first simplify the inner cross product vector term, $$\left(\vec{a} + \vec{b}\right) \times \left(\vec{a} \times \vec{b}\right)$$, by distributing it:
$$\left(\vec{a} + \vec{b}\right) \times \left(\vec{a} \times \vec{b}\right) = \vec{a} \times \left(\vec{a} \times \vec{b}\right) + \vec{b} \times \left(\vec{a} \times \vec{b}\right)$$
Using the vector triple product expansion formula $$\vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z})\vec{y} - (\vec{x} \cdot \vec{y})\vec{z}$$:
- $$\vec{a} \times \left(\vec{a} \times \vec{b}\right) = (\vec{a} \cdot \vec{b})\vec{a} - |\vec{a}|^2\vec{b}$$
- $$\vec{b} \times \left(\vec{a} \times \vec{b}\right) = -\vec{b} \times \left(\vec{b} \times \vec{a}\right) = -\left[ (\vec{b} \cdot \vec{a})\vec{b} - |\vec{b}|^2\vec{a} \right] = |\vec{b}|^2\vec{a} - (\vec{a} \cdot \vec{b})\vec{b}$$
Adding these two results together:
$$\left(\vec{a} + \vec{b}\right) \times \left(\vec{a} \times \vec{b}\right) = \left( \vec{a} \cdot \vec{b} + |\vec{b}|^2 \right)\vec{a} - \left( |\vec{a}|^2 + \vec{a} \cdot \vec{b} \right)\vec{b}$$
Now, we compute the final outer cross product with $(\vec{a} - \vec{b})$:
$$\left[ \left( \vec{a} \cdot \vec{b} + |\vec{b}|^2 \right)\vec{a} - \left( |\vec{a}|^2 + \vec{a} \cdot \vec{b} \right)\vec{b} \right] \times (\vec{a} - \vec{b})$$
Since $$\vec{a} \times \vec{a} = 0$$ and $$\vec{b} \times \vec{b} = 0$$, this distributes cleanly into:
$$= -\left( \vec{a} \cdot \vec{b} + |\vec{b}|^2 \right)(\vec{a} \times \vec{b}) - \left( |\vec{a}|^2 + \vec{a} \cdot \vec{b} \right)(\vec{b} \times \vec{a})$$ $$= -\left( \vec{a} \cdot \vec{b} + |\vec{b}|^2 \right)(\vec{a} \times \vec{b}) + \left( |\vec{a}|^2 + \vec{a} \cdot \vec{b} \right)(\vec{a} \times \vec{b})$$ $$= \left( |\vec{a}|^2 - |\vec{b}|^2 \right) (\vec{a} \times \vec{b})$$
Step 2: Determine the value of $$\lambda$$
Now let's compute $$|\vec{a}|^2$$ and $$|\vec{b}|^2$$ from our given vectors:
- $$|\vec{a}|^2 = \lambda^2 + 3^2 + (-1)^2 = \lambda^2 + 10$$
- $$|\vec{b}|^2 = 2^2 + \lambda^2 + 1^2 = \lambda^2 + 5$$
Subtracting them gives a constant:
$$|\vec{a}|^2 - |\vec{b}|^2 = (\lambda^2 + 10) - (\lambda^2 + 5) = 5$$
Substitute this back into our simplified relation:
$$5(\vec{a} \times \vec{b}) = 25\hat{i} - 20\hat{j} - 10\hat{k}$$ $$\vec{a} \times \vec{b} = 5\hat{i} - 4\hat{j} - 2\hat{k}$$
Let's verify this using the cross-product determinant of $$\vec{a}$$ and $$\vec{b}$$:
$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \lambda & 3 & -1 \\ 2 & \lambda & 1 \end{vmatrix} = (3 + \lambda)\hat{i} - (\lambda + 2)\hat{j} + (\lambda^2 - 6)\hat{k}$$
Comparing components with $$5\hat{i} - 4\hat{j} - 2\hat{k}$$:
- $$3 + \lambda = 5 \implies \lambda = 2$$
- $$-(\lambda + 2) = -4 \implies \lambda = 2$$
Both yield $$\lambda = 2$$.
Step 3: Find the target expression value
We need to find $$\left| \lambda\left(\vec{a} + \vec{b}\right) \times \left(\vec{a} - \vec{b}\right) \right|^2$$.
Expanding the cross product inside the norm:
$$\left(\vec{a} + \vec{b}\right) \times \left(\vec{a} - \vec{b}\right) = \vec{a}\times\vec{a} - \vec{a}\times\vec{b} + \vec{b}\times\vec{a} - \vec{b}\times\vec{b} = -2(\vec{a} \times \vec{b})$$
Substitute this into the expression:
$$\left| -2\lambda (\vec{a} \times \vec{b}) \right|^2 = 4\lambda^2 |\vec{a} \times \vec{b}|^2$$
Using $$\lambda = 2$$ and $$|\vec{a} \times \vec{b}|^2 = 5^2 + (-4)^2 + (-2)^2 = 25 + 16 + 4 = 45$$:
$$\text{Value} = 4(2)^2 \cdot 45 = 16 \cdot 45 = 720$$
Correct Option: (3) 720
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