Question: Find $ y $ such that the vectors $ \begin{pmatrix} 2 \\ y \\ -1 \end{pmatrix} $ and $ \begin{pmatrix} 3 \\ -2 \\ y \end{pmatrix} $ are orthogonal.

["Title: How to Find $ y $ When Two Vectors Are Orthogonal: A Step-by-Step Guide", "In linear algebra, one essential concept is orthogonality—two vectors are orthogonal when their dot product equals zero. Understanding how to find the unknown variable $ y $ that makes two vectors orthogonal is valuable in fields like computer graphics, engineering, data science, and more.", "In this SEO-optimized article, we’ll explore how to determine $ y $ such that the vectors", "$$\n\begin{pmatrix} 2 \ y \ -1 \end{pmatrix} \quad \ ext{and} \quad \begin{pmatrix} 3 \ -2 \ y \end{pmatrix}\n$$", "are orthogonal. We’ll break down the process clearly, include relevant keywords, and explain why this concept matters.", "---", "### What Does It Mean for Vectors to Be Orthogonal?", "Two vectors are orthogonal if their dot product is zero. The dot product of two vectors $ \begin{pmatrix} a_1 \ a_2 \ a_3 \end{pmatrix} $ and $ \begin{pmatrix} b_1 \ b_2 \ b_3 \end{pmatrix} $ is computed as:", "$$\n\vec{u} \cdot \vec{v} = a_1b_1 + a_2b_2 + a_3b_3\n$$", "Using this definition, let’s compute the dot product of the given vectors.", "---", "### Step 1: Compute the Dot Product", "Let:", "- $ \vec{u} = \begin{pmatrix} 2 \ y \ -1 \end{pmatrix} $\n- $ \vec{v} = \begin{pmatrix} 3 \ -2 \ y \end{pmatrix} $", "Then:", "$$\n\vec{u} \cdot \vec{v} = (2)(3) + (y)(-2) + (-1)(y) = 6 - 2y - y = 6 - 3y\n$$", "---", "### Step 2: Set the Dot Product Equal to Zero", "For orthogonality:", "$$\n6 - 3y = 0\n$$", "Solving for $ y $:", "$$\n3y = 6 \quad \Rightarrow \quad y = 2\n$$", "---", "### Step 3: Verify the Solution", "Plug $ y = 2 $ back into the vectors:", "- $ \vec{u} = \begin{pmatrix} 2 \ 2 \ -1 \end{pmatrix} $\n- $ \vec{v} = \begin{pmatrix} 3 \ -2 \ 2 \end{pmatrix} $", "Dot product:", "$$\n2(3) + 2(-2) + (-1)(2) = 6 - 4 - 2 = 0\n$$", "Confirmed—vectors are indeed orthogonal when $ y = 2 $.", "---", "### Why Orthogonality Matters (SEO Keywords)", "Understanding orthogonal vectors enhances your ability to work with:", "- Linear transformations (e.g., projections)\n- Eigenvalues and eigenvectors\n- Machine learning algorithms (PCA, SVM)\n- 3D graphics and spatial computing\n- Signal processing and Fourier analysis", "By mastering the dot product method shown here, you build a foundation for advanced mathematical and computational applications.", "---", "### Final Answer", "$$\n\boxed{y = 2}\n$$", "Finding $ y $ such that the vectors are orthogonal reduces to solving a simple equation derived from the dot product: $ 6 - 3y = 0 \Rightarrow y = 2 $. This method applies broadly to any 3D vector pair and is a cornerstone in vector analysis.", "---", "Keywords: orthogonal vectors, dot product, find y orthogonal vector, linear algebra, vector dot product, education, math tutorial, coordinate geometry, 3D vectors, vector analysis, projection, mathematical concepts.", "---", "Meta Description: Learn how to find $ y $ so that two vectors are orthogonal by computing their dot product. Step-by-step solution, verification, and real-world applications in math and STEM fields."]