Solution: Two vectors are orthogonal if their dot product is zero. Compute the dot product: $ x \cdot 3 + 2 \cdot (-x) = 3x - 2x = x $. Set $ x = 0 $. Thus, $ oxed{0} $ is the solution.

["Understanding Orthogonal Vectors: When Do Two Vectors Truly Quit Everything?", "When studying vector geometry, a fundamental concept academic and application-driven fields rely on is orthogonality. But what does it really mean when two vectors are orthogonal? The key lies in a powerful mathematical shortcut: their dot product must equal zero.", "### What Are Orthogonal Vectors?", "In vector mathematics, two vectors are orthogonal (or perpendicular) if they meet at a right angle. This geometric interpretation translates seamlessly into algebraic form via the dot product. If vectors a and b are orthogonal, one simple criterion is:", "> Their dot product is zero.", "Mathematically, this means:\n$$\n\mathbf{a} \cdot \mathbf{b} = 0\n$$", "This principle simplifies orthogonality checks without needing angle measurements, making computations efficient and intuitive.", "### A Clear Example: Compute the Dot Product", "Consider two vectors expressed in terms of a variable $ x $:", "$$\n\mathbf{a} = (x,\ 3) \quad \ ext{and} \quad \mathbf{b} = (3,\ -x)\n$$", "To determine orthogonality, compute their dot product directly:", "$$\n\mathbf{a} \cdot \mathbf{b} = x \cdot 3 + 2 \cdot (-x) = 3x - 2x = x\n$$", "Now, for orthogonality, set the dot product equal to zero:", "$$\nx = 0\n$$", "Thus, the only value of $ x $ that makes the vectors perpendicular is $ \boxed{0} $.", "### Why This Matters", "Identifying orthogonal vectors using the dot product is essential in many real-world applications—from computer graphics and machine learning to signal processing and physics. A zero dot product signals no component along each other, confirming independence in directional terms.", "### Final Takeaway", "Orthogonality is both a geometric intuition and an algebraic test: when the dot product of two vectors equals zero, they are orthogonal. In our example, solving $ x = 0 $ reveals that only at $ x = 0 $ do these vectors form a right angle.", "---", "Understanding this core principle empowers precise problem-solving across mathematics and technology. Remember: zero dot product = orthogonality."]