1. The problem is to find the value of the vector expression $2\mathbf{i} \times 2\mathbf{j} \cdot \mathbf{k}$. 2. Recall the vector operations involved: - The cross product $\times$ between two vectors results in a vector perpendicular to both. - The dot product $\cdot$ between two vectors results in a scalar. 3. First, compute the cross product $2\mathbf{i} \times 2\mathbf{j}$: $$2\mathbf{i} \times 2\mathbf{j} = 4 (\mathbf{i} \times \mathbf{j})$$ 4. Using the right-hand rule and standard unit vectors: $$\mathbf{i} \times \mathbf{j} = \mathbf{k}$$ 5. Therefore: $$4 (\mathbf{i} \times \mathbf{j}) = 4\mathbf{k}$$ 6. Now compute the dot product of this result with $\mathbf{k}$: $$4\mathbf{k} \cdot \mathbf{k} = 4 (\mathbf{k} \cdot \mathbf{k})$$ 7. Since the dot product of a unit vector with itself is 1: $$\mathbf{k} \cdot \mathbf{k} = 1$$ 8. Hence: $$4 (\mathbf{k} \cdot \mathbf{k}) = 4 \times 1 = 4$$ 9. Final answer: $$\boxed{4}$$