1. We are given the structural vector equation:
$$\vec{x} + \vec{a} \times \vec{x} = \vec{b}$$
2. Take the scalar dot product of both sides of the equation with vector $$\vec{a}$$:
$$\vec{a} \cdot \vec{x} + \vec{a} \cdot (\vec{a} \times \vec{x}) = \vec{a} \cdot \vec{b}$$
Since the vectors $$\vec{a}$$ and $$\vec{b}$$ are given to be orthogonal, $$\vec{a} \cdot \vec{b} = 0$$. Additionally, any scalar triple product with a repeated vector is zero ($$\vec{a} \cdot (\vec{a} \times \vec{x}) = 0$$). This simplifies to:
$$\vec{a} \cdot \vec{x} = 0$$
3. Now, find the squared magnitude of both sides of the original vector equation:
$$|\vec{x} + \vec{a} \times \vec{x}|^2 = |\vec{b}|^2$$
4. Expand the left-hand side using the vector dot product expansion rules:
$$|\vec{x}|^2 + |\vec{a} \times \vec{x}|^2 + 2 \cdot \vec{x} \cdot (\vec{a} \times \vec{x}) = |\vec{b}|^2$$
5. Apply vector identities to simplify the terms:
* The scalar triple product $$\vec{x} \cdot (\vec{a} \times \vec{x})$$ contains a repeated vector, so it evaluates to 0.
* Use Lagrange's identity for the cross-product magnitude:
$$|\vec{a} \times \vec{x}|^2 = |\vec{a}|^2|\vec{x}|^2 - (\vec{a} \cdot \vec{x})^2$$
6. Substitute these identity simplifications back into the expanded magnitude equation:
$$|\vec{x}|^2 + \left( |\vec{a}|^2|\vec{x}|^2 - (\vec{a} \cdot \vec{x})^2 \right) + 0 = |\vec{b}|^2$$
7. Substitute the known constant parameters ($$|\vec{a}| = 1$$, $$|\vec{b}| = 2 \implies |\vec{b}|^2 = 4$$, and our derived step $$\vec{a} \cdot \vec{x} = 0$$):
$$|\vec{x}|^2 + \left( (1)^2|\vec{x}|^2 - 0^2 \right) = 4$$
$$|\vec{x}|^2 + |\vec{x}|^2 = 4 \implies 2|\vec{x}|^2 = 4$$
$$|\vec{x}|^2 = 2$$
8. Calculate the final requested numerical value:
$$10|\vec{x}|^2 = 10 \cdot 2 = 20$$
91- 6303239042
