Solution: The diameter of the semicircular drum is 12 meters, so the radius is half the diameter. Thus, $ r = \frac{12}{2} = 6 $ meters. The radius is $ \boxed{6} $ meters.Question: Find $ x $ such that the vectors $ \begin{pmatrix} x \\ 3 \end{pmatrix} $ and $ \begin{pmatrix} 2 \\ -x \end{pmatrix} $ are orthogonal.

["Finding $ x $ so That Vectors Are Orthogonal: A Step-by-Step Guide", "Undirected vectors play a crucial role in physics, engineering, and computer graphics—especially when ensuring movements, forces, or shapes interact correctly. One key relationship is when vectors are orthogonal, meaning they meet at right angles. In this article, we’ll explore how to find the value of $ x $ that makes two vectors orthogonal, using a clear geometric principle and basic algebra.", "### The Concept of Orthogonal Vectors", "Two vectors are orthogonal if their dot product equals zero. For two vectors $ \vec{u} = \begin{pmatrix} a \ b \end{pmatrix} $ and $ \vec{v} = \begin{pmatrix} c \ d \end{pmatrix} $, the dot product is computed as:\n$$\n\vec{u} \cdot \vec{v} = ac + bd\n$$\nSetting the dot product equal to zero gives the condition for orthogonality:\n$$\nac + bd = 0\n$$", "### Applying It to the Given Vectors", "We are given:\n$$\n\vec{u} = \begin{pmatrix} x \ 3 \end{pmatrix}, \quad \vec{v} = \begin{pmatrix} 2 \ -x \end{pmatrix}\n$$\nTo find $ x $ such that $ \vec{u} $ and $ \vec{v} $ are orthogonal, compute their dot product:\n$$\nx \cdot 2 + 3 \cdot (-x) = 0\n$$\nSimplify:\n$$\n2x - 3x = 0\n$$\n$$\n- x = 0\n$$\n$$\nx = 0\n$$", "### Verifying the Solution", "Plugging $ x = 0 $ back into the original vectors:\n$$\n\vec{u} = \begin{pmatrix} 0 \ 3 \end{pmatrix}, \quad \vec{v} = \begin{pmatrix} 2 \ 0 \end{pmatrix}\n$$\nTheir dot product is:\n$$\n0 \cdot 2 + 3 \cdot 0 = 0\n$$\nSince the dot product is zero, the vectors are indeed orthogonal—confirming our solution.", "### Conclusion", "The value of $ x $ that makes the vectors $ \begin{pmatrix} x \ 3 \end{pmatrix} $ and $ \begin{pmatrix} 2 \ -x \end{pmatrix} $ orthogonal is $ \boxed{0} $. Understanding this fundamental property helps in solving applications across disciplines, from ensuring perpendicularity in design to calculating forces in equilibrium.", "---\nKeywords: orthogonal vectors, dot product, find x, perpendicular vectors, vector geometry, linear algebra basics\nSchema: Understanding Orthogonal Vectors – Finding x Using Dot Product"]