Solution:** To find a vector \(\mathbf{v}\) that is perpendicular to \(\mathbf{w} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\), we need \(\mathbf{v} \cdot \mathbf{w} = 0\). Let \(\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}\). The dot product condition gives:

["Finding a Perpendicular Vector to (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}): The Mathematical Solution", "When working with vectors in 2D space, one fundamental concept is identifying vectors that are perpendicular (orthogonal) to a given vector. In this article, we explore the solution to the problem of finding a vector (\mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix}) that is perpendicular to (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}), based on the dot product condition.", "### Understanding Vector Perpendicularity", "Two vectors are perpendicular if their dot product equals zero. Mathematically, for vectors (\mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix}) and (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}), the condition is:", "[\n\mathbf{v} \cdot \mathbf{w} = 0\n]", "The dot product is computed as:", "[\n\mathbf{v} \cdot \mathbf{w} = x \cdot 3 + y \cdot 4 = 3x + 4y\n]", "Setting this equal to zero gives the equation:", "[\n3x + 4y = 0\n]", "### Solving for a Perpendicular Vector", "This linear equation has infinitely many solutions, because if one component determines the other, we can express one variable in terms of the other.", "Solve for (y) in terms of (x):", "[\n4y = -3x \implies y = -\frac{3}{4}x\n]", "Thus, any vector of the form:", "[\n\mathbf{v} = \begin{bmatrix} x \ -\frac{3}{4}x \end{bmatrix}\n]", "is perpendicular to (\mathbf{w}). To avoid fractions and simplify, we can choose any nonzero value for (x). For example, let (x = 4). Then:", "[\ny = -\frac{3}{4}(4) = -3\n]", "So one simple solution vector is:", "[\n\mathbf{v} = \begin{bmatrix} 4 \ -3 \end{bmatrix}\n]", "We can verify that:", "[\n\mathbf{v} \cdot \mathbf{w} = 3(4) + 4(-3) = 12 - 12 = 0\n]", "Confirming orthogonality.", "### Why This Works", "The direction of (\mathbf{w} = [3, 4]) defines a line with slope (\frac{4}{3}). A perpendicular vector must have a slope that is the negative reciprocal, namely (-\frac{3}{4})—exactly the slope of (\mathbf{v}) above.", "### General Form of Perpendicular Vectors", "All perpendicular vectors to (\mathbf{w}) lie along the line spanned by (\mathbf{v} = \begin{bmatrix} 4 \ -3 \end{bmatrix}), or any scalar multiple of it. Thus, the solution space is one-dimensional and represents all orthogonal directions to (\mathbf{w}).", "### Summary", "To find a vector (\mathbf{v}) perpendicular to (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}):", "- Set up the dot product condition: (3x + 4y = 0).\n- Solve to express (y) in terms of (x), yielding (\mathbf{v} = \begin{bmatrix} x \ -\frac{3}{4}x \end{bmatrix}).\n- Pick any nonzero (x) for a non-trivial solution; (\begin{bmatrix} 4 \ -3 \end{bmatrix}) is a simple solution.\n- Verify by computing the dot product.", "This method leverages the core property of dot products to identify orthogonality, providing a clear and efficient solution in vector algebra and applications such as computer graphics, physics, and machine learning.", "---", "Keywords: perpendicular vector, dot product, vector algebra, orthogonal vector, solution to ( \mathbf{v} \cdot \mathbf{w} = 0 ), (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}), find vector (\mathbf{v}), linear algebra, math solutions."]