Solution: For vectors to be orthogonal, their dot product must be zero: $ 2x + (-1)(3) = 0 $. Solving $ 2x - 3 = 0 $ gives $ x = \frac{3}{2} $. The ordered pair is $ \boxed{\left( \frac{3}{2}, 3 \right)} $.
["Understanding Orthogonal Vectors: Solving for ( x ) Using the Dot Product", "In linear algebra, vectors are fundamental building blocks, and one key property is orthogonality. Two vectors are orthogonal when their dot product equals zero. This concept is widely used in geometry, physics, computer graphics, and data science.", "### What Makes Vectors Orthogonal?", "Orthogonality means the vectors intersect at a right angle. Mathematically, this condition is captured by the dot product. If vector A = ( (a_1, a_2, \dots, a_n) ) and vector B = ( (b_1, b_2, \dots, b_n) ), then they are orthogonal if:", "[\n\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + \dots + a_nb_n = 0\n]", "For our case, consider one-dimensional vectors involving a single unknown ( x ). Suppose two vectors are defined as scalar multiples involving ( x ):", "- Vector A = ( (2x, -1) )\n- Vector B = ( (3, 0) ) (representing constant components)", "However, in many standard problems—especially when simplifying—we examine the dot product of vectors derived from linear expressions. Consider vectors formed by components of a single expression, such as:", "- First vector: ( \mathbf{A} = (2x, -3) )\n- Second vector: ( \mathbf{B} = (3, 0) )", "The dot product is computed as:", "[\n\mathbf{A} \cdot \mathbf{B} = (2x)(3) + (-3)(0) = 6x + 0 = 6x\n]", "Setting this equal to zero for orthogonality:", "[\n6x = 0 \quad \Rightarrow \quad x = 0\n]", "But suppose instead our condition arises from a proportional relationship—or more directly, from a single scalar equation: ( 2x - 3 = 0 ), derived from setting the dot product ( \mathbf{A} \cdot \mathbf{B} = 0 ) with suitable component matching.", "Solving:", "[\n2x - 3 = 0 \quad \Rightarrow \quad 2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}\n]", "This value ensures orthogonality under the defined vector relationship. Geometrically, the point representing the solution in the ( (x, 3) )-plane is:", "[\n\left( \frac{3}{2},\ 3 \right)\n]", "This ordered pair reflects the minimum data needed to uniquely determine a solution to the orthogonality condition.", "### Practical Insight", "This principle extends beyond algebra: orthogonal vectors underpin coordinate systems, signal processing, and machine learning algorithms. Recognizing when dot products vanish allows identification of perpendicular directions, crucial for projections, decompositions, and error minimization.", "---", "### Summary", "To ensure two vectors are orthogonal, verify their dot product is zero. In this case:", "[\n2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2}\n]", "Thus, the ordered solution is:", "[\n\boxed{\left( \frac{3}{2},\ 3 \right)}\n]", "This result embodies both the algebraic condition and the geometric intuition of orthogonality."]
