Question: Find the value of \(x\) such that the vectors \(\begin{pmatrix} x \\ 2 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}\) are orthogonal.
["Find the Value of ( x ) That Makes the Vectors Orthogonal", "When working with vectors in three-dimensional space, one important concept is orthogonality — when two vectors are perpendicular to each other. This means their dot product equals zero. In this article, we’ll solve the problem: Find the value of ( x ) such that the vectors ( \begin{pmatrix} x \ 2 \ -1 \end{pmatrix} ) and ( \begin{pmatrix} 3 \ -1 \ 4 \end{pmatrix} ) are orthogonal.", "### What Are Orthogonal Vectors?", "Two vectors are orthogonal if their dot product equals zero. The dot product of two vectors ( \mathbf{u} = \begin{pmatrix} u_1 \ u_2 \ u_3 \end{pmatrix} ) and ( \mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix} ) is calculated as:", "[\n\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3\n]", "### Step-by-Step: Setting Up the Equation", "Given the vectors:\n[\n\begin{pmatrix} x \ 2 \ -1 \end{pmatrix} \quad \ ext{and} \quad \begin{pmatrix} 3 \ -1 \ 4 \end{pmatrix}\n]", "Compute their dot product:", "[\nx \cdot 3 + 2 \cdot (-1) + (-1) \cdot 4 = 0\n]", "Simplify:", "[\n3x - 2 - 4 = 0\n]", "Combine like terms:", "[\n3x - 6 = 0\n]", "### Solve for ( x )", "Add 6 to both sides:", "[\n3x = 6\n]", "Divide both sides by 3:", "[\nx = 2\n]", "### Conclusion", "The value of ( x ) that makes the vectors ( \begin{pmatrix} x \ 2 \ -1 \end{pmatrix} ) and ( \begin{pmatrix} 3 \ -1 \ 4 \end{pmatrix} ) orthogonal is:", "[\n\boxed{2}\n]", "This result confirms that when ( x = 2 ), the vectors are perpendicular, satisfying the geometric condition of orthogonality.", "### Bonus: Why Orthogonality Matters", "Orthogonal vectors are fundamental in linear algebra and have key applications in fields like computer graphics, signal processing, and machine learning. Understanding how to find such values helps in constructing coordinate systems, optimizing algorithms, and simplifying complex problems.", "---", "Keywords: orthogonal vectors, dot product, find ( x ), vector algebra, 3D vectors, linear algebra, mathematical solutions"]
