Solution: Two vectors are orthogonal if their dot product is zero: $ x \cdot 3 + 2 \cdot (-1) = 0 $. Simplify: $ 3x - 2 = 0 \Rightarrow x = \frac{2}{3} $.

["Understanding Orthogonality in Vectors: A Simple Explanation with Equation Examples", "In linear algebra, the concept of orthogonal vectors is fundamental to geometry, physics, and engineering. Two vectors are considered orthogonal if their dot product equals zero. This mathematical condition reveals geometric insight: the vectors are perpendicular to each other.", "Consider the equation expressing orthogonality through a dot product:\n$$\nx \cdot 3 + 2 \cdot (-1) = 0\n$$", "This expression models the dot product of two vectors — one scalar-coefficiented and the other involving a constant — often written in one-dimensional form for educational clarity. By simplifying, we apply the standard dot product rule:", "$$\n3x - 2 = 0\n$$", "Solving this linear equation is straightforward:", "$$\n3x = 2 \quad \Rightarrow \quad x = \frac{2}{3}\n$$", "This value of $ x = \frac{2}{3} $ ensures that the two vectors are orthogonal. Geometrically, this means if one vector represents a scalar multiple of 3 and the other a constant value of -2, their alignment at right angles satisfies this mathematical constraint.", "Mastering orthogonality through equations like $ 3x - 2 = 0 $ not only strengthens algebraic skills but also deepens understanding of vector geometry. Whether in physics, computer graphics, or machine learning, recognizing orthogonal relationships helps optimize systems, improve accuracy, and clarify spatial concepts.", "In summary, if $ x \cdot 3 + 2 \cdot (-1) = 0 $, then solving $ 3x - 2 = 0 $ gives $ x = \frac{2}{3} $, demonstrating how a simple equation encodes a key geometric principle: orthogonality through a zero dot product."]