Solution: Use the Cauchy-Schwarz inequality: $(2^2 + 3^2 + 4^2)(x^2 + y^2 + z^2) \geq (2x + 3y + 4z)^2$. This gives $29(x^2 + y^2 + z^2) \geq 144$, so $x^2 + y^2 + z^2 \geq rac{144}{29}$. Equality holds when $rac{x}{2} = rac{y}{3} = rac{z}{4} = k$, leading to $x = 2k$, $y = 3k$, $z = 4k$. Substituting into $2x + 3y + 4z = 12$ gives $4k + 9k + 16k = 29k = 12$, so $k = rac{12}{29}$. Thus, the minimum value is $oxed{\dfrac{144}{29}}$.

["Maximizing Efficiency with the Cauchy-Schwarz Inequality: Deriving the Minimum Value", "Optimization problem-solving is a cornerstone of mathematical analysis, particularly when leveraging powerful inequalities like the Cauchy-Schwarz inequality. This elegant inequality not only provides insight into vector relationships but also enables us to derive precise bounds in algebraic expressions—an essential skill in both theoretical and applied mathematics.", "### The Cauchy-Schwarz Inequality in Action", "Consider the inequality:", "[\n(2^2 + 3^2 + 4^2)(x^2 + y^2 + z^2) \geq (2x + 3y + 4z)^2\n]", "Start by computing the left-hand sum of squares:", "[\n2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29\n]", "Then the inequality becomes:", "[\n29(x^2 + y^2 + z^2) \geq (2x + 3y + 4z)^2\n]", "Now apply the fundamental insight that for real numbers, a minimum can be found by bounding the right-hand side:", "[\n(2x + 3y + 4z)^2 \leq 144 \quad \ ext{when equality is approached}\n]", "This follows because given $2x + 3y + 4z = 12$, the upper bound arises from maximizing the dot product under fixed vector lengths—an application closely tied to vector geometry.", "Combining both results:", "[\n29(x^2 + y^2 + z^2) \geq 144\n]", "Dividing both sides by 29 yields:", "[\nx^2 + y^2 + z^2 \geq \frac{144}{29}\n]", "### Equality Condition and the Optimal Vector Ratio", "Equality holds in Cauchy-Schwarz if and only if the vectors are proportional. That is, there exists a constant $k$ such that:", "[\n\frac{x}{2} = \frac{y}{3} = \frac{z}{4} = k\n]", "Substitute $x = 2k$, $y = 3k$, $z = 4k$ into the constraint $2x + 3y + 4z = 12$:", "[\n2(2k) + 3(3k) + 4(4k) = 4k + 9k + 16k = 29k = 12\n]", "Solving for $k$:", "[\nk = \frac{12}{29}\n]", "Then compute the minimum value of $x^2 + y^2 + z^2$:", "[\nx^2 + y^2 + z^2 = (2k)^2 + (3k)^2 + (4k)^2 = 4k^2 + 9k^2 + 16k^2 = 29k^2\n]", "Substitute $k = \frac{12}{29}$:", "[\nx^2 + y^2 + z^2 = 29 \left( \frac{12}{29} \right)^2 = 29 \cdot \frac{144}{841} = \frac{144}{29}\n]", "### Conclusion: A Precise Minimum Through Symmetry", "The minimum value of $x^2 + y^2 + z^2$, subject to the constraint $2x + 3y + 4z = 12$, is:", "[\n\boxed{\dfrac{144}{29}}\n]", "This result exemplifies how the Cauchy-Schwarz inequality transforms otherwise complex optimization into a clean geometric proportionality problem. Whether used in algebra, physics, or machine learning, understanding such bounds empowers smarter, more efficient solutions."]