Solution: Two vectors are orthogonal if their dot product is zero. Compute the dot product: \(3x + (2)(-1) + (-1)(4) = 3x - 2 - 4 = 3x - 6\). Set this equal to zero: \(3x - 6 = 0 \implies x = 2\).
["Understanding Orthogonal Vectors Through Dot Products: A Step-by-Step Solution", "When studying vectors in mathematics, a key concept is orthogonality—when two vectors are perpendicular to each other. A powerful criterion to determine orthogonality is based on the dot product: two vectors are orthogonal if and only if their dot product equals zero.", "### What Does It Mean for Two Vectors to Be Orthogonal?", "Orthogonal vectors interact mathematically in a special way: their dot product is zero. This condition arises naturally in geometry and physics, where perpendicularity signals independence in direction without overlap.", "### Example: Solving for Orthogonality via Dot Product", "Consider two linear expressions representing vector components:", "[\n3x + (2)(-1) + (-1)(4)\n]", "First, simplify the constant terms:", "[\n3x + (2)(-1) + (-1)(4) = 3x - 2 - 4 = 3x - 6\n]", "This expression is actually the dot product of two vectors. Let’s interpret this vectorually.", "#### Step 1: Represent the Vectors\nSuppose:\n[\n\mathbf{u} = \langle 3x, -2, -4 \rangle \quad \ ext{and} \quad \mathbf{v} = \langle 1, 1, 1 \rangle\n]", "Their dot product is computed as:\n[\n\mathbf{u} \cdot \mathbf{v} = (3x)(1) + (-2)(1) + (-4)(1) = 3x - 2 - 4 = 3x - 6\n]", "#### Step 2: Set the Dot Product to Zero\nFor orthogonality:\n[\n3x - 6 = 0\n]", "Solve for (x):\n[\n3x = 6 \implies x = 2\n]", "Thus, the vectors are orthogonal only when (x = 2).", "### Why This Matters", "This calculation demonstrates how algebraic manipulation connects to geometric insight—finding when vectors intersect at right angles. It applies broadly in computer graphics, physics, engineering, and machine learning, where orthogonal components enable efficient decomposition and analysis.", "### Summary", "- Orthogonal vectors have a dot product of zero.\n- Solving (3x - 6 = 0) yields (x = 2), the precise value ensuring orthogonality.\n- This method is a clean, generalizable tool for checking vector relationships.", "By mastering this principle, you unlock deeper geometric reasoning and strengthen your foundation in linear algebra.", "---", "Keywords: orthogonal vectors, dot product, geometric interpretation, vector mathematics, perpendicular vectors, mathematical solution, linear algebra, coordinate geometry"]
