Solution: For orthogonality, their dot product must be zero: $1 \cdot x + x \cdot 3 + 2 \cdot (-1) = 0$. Simplifying, $x + 3x - 2 = 0 \Rightarrow 4x = 2 \Rightarrow x = \frac{1}{2}$.
["Understanding Orthogonality: How to Solve for x Using Dot Product Conditions", "In mathematics—particularly in linear algebra and geometry—orthogonality is a fundamental concept that describes vectors being perpendicular to one another. A key algebraic tool to determine orthogonality is the dot product: if two vectors are orthogonal, their dot product equals zero. This principle offers a clear, straightforward way to solve problems involving perpendicular vectors.", "---", "### The Dot Product Condition for Orthogonality", "For two vectors represented by their components, orthogonality is expressed as:", "[\n\vec{v} \cdot \vec{w} = 0\n]", "That is, the sum of the products of their corresponding components must equal zero. In simple terms, if vector A = ([a_1, a_2, \dots]) and vector B = ([b_1, b_2, \dots]), then orthogonality requires:", "[\na_1b_1 + a_2b_2 + \cdots = 0\n]", "---", "### Applying This Concept: A Real-World Equation", "Consider the following dot product equation modeling orthogonality:", "[\n1 \cdot x + x \cdot 3 + 2 \cdot (-1) = 0\n]", "This expression combines three terms involving the unknown variable (x), each representing a component product. Let’s break it down step by step.", "- The first term: (1 \cdot x = x)\n- The second term: (x \cdot 3 = 3x)\n- The third term: (2 \cdot (-1) = -2)", "Putting it all together:", "[\nx + 3x - 2 = 0\n]", "---", "### Simplifying the Equation", "Combine like terms:", "[\n(1x + 3x) - 2 = 4x - 2 = 0\n]", "Now solve for (x):", "[\n4x = 2\n]", "[\nx = \frac{2}{4} = \frac{1}{2}\n]", "---", "### Why This Solution Works", "When (x = \frac{1}{2}), the dot product becomes:", "[\n1 \cdot \frac{1}{2} + \frac{1}{2} \cdot 3 + 2 \cdot (-1) = \frac{1}{2} + \frac{3}{2} - 2 = \frac{4}{2} - 2 = 2 - 2 = 0\n]", "This confirms orthogonality—because the dot product is zero, the vectors (modeled by these components) are perpendicular.", "---", "### Conclusion", "The approach of setting dot products to zero is a powerful and elegant method to solve orthogonality problems. It transforms geometric intuition into a simple algebraic equation that is easy to solve. Understanding this process not only helps in algebra and linear algebra but also strengthens your ability to work with vectors in physics, computer graphics, and engineering.", "Key Takeaways:\n- Orthogonal vectors have a dot product of zero.\n- Break down vector expressions carefully and combine terms.\n- Solving linear equations is a core skill in vector math.", "Master orthogonality with dot products—it’s the foundation for countless applications in math and science!", "---", "Keywords: orthogonality, dot product, vector math, solving linear equations, geometry, linear algebra tutorial, perpendicular vectors, 4x = 2, simplify dot product, mathematical application"]
