Since \(\|\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}\| \geq 0\), the projection of \(\mathbf{u}\) onto the plane perpendicular to \(\mathbf{v}\) satisfies:
["Understanding the Projection of a Vector onto a Plane Using Inner Products – Why the Norm is Non-Negative", "Since (|\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}| \geq 0), it follows that the projection of the vector (\mathbf{u}) onto the plane perpendicular to (\mathbf{v}) is well-defined and yields the shortest Euclidean distance from (\mathbf{u}) to the plane. This fundamental result in linear algebra and geometry underpins projections in machine learning, signal processing, and optimization.", "---", "### The Projection onto the Perpendicular Plane", "Given any vector (\mathbf{u}) in (\mathbb{R}^n) and a nonzero vector (\mathbf{v}), the vector (\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}) lies in the plane perpendicular to (\mathbf{v}). This residual vector represents the component of (\mathbf{u}) orthogonal to (\mathbf{v})—precisely the projection of (\mathbf{u}) onto the subspace perpendicular to (\mathbf{v}).", "#### Mathematical Justification", "The norm squared of this residual vector is:", "[\n|\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}|^{\sn]a} = \mathbf{u} \cdot \mathbf{u} - 2 (\mathbf{u} \cdot \mathbf{v})^2 + (\mathbf{u} \cdot \mathbf{v})^2 = |\mathbf{u}|^2 - (\mathbf{u} \cdot \mathbf{v})^2.\n]", "Because the inner product is positive semidefinite—namely, ((\mathbf{a} \cdot \mathbf{b})^2 \leq |\mathbf{a}|^2 |\mathbf{b}|^2)—it follows that:", "[\n|\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}|^{\sn]a} = |\mathbf{u}|^2 - (\mathbf{u} \cdot \mathbf{v})^2 \geq 0.\n]", "This non-negativity guarantees that the quantity is real and non-negative, validating (\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}) as a valid vector orthogonal to (\mathbf{v}).", "---", "### The Perpendicular Projection Formula", "The projection (\mathbf{u}{\perp}) of (\mathbf{u}) onto the plane perpendicular to (\mathbf{v}) is:", "[\n\mathbf{u}} = \mathbf{u} - \mathrm{proj{\mathbf{v}} \mathbf{u} = \mathbf{u} - \left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \right) \mathbf{v}.\n]", "This expression subtracts the component of (\mathbf{u}) in the direction of (\mathbf{v}), leaving only the orthogonal component.", "---", "### Geometric Interpretation", "Geometrically, (\mathbf{u}) is the projection, confirming the optimality of the formula derived from minimizing a quadratic functional via inner products.", "---", "### Practical Implications", "- }) lies in the hyperplane orthogonal to (\mathbf{v}), minimizing the distance (|\mathbf{u} - \mathbf{w}|_{|\cdot|}) over all vectors (\mathbf{w}) in that plane. This minimum distance is achieved uniquely when (\mathbf{wMachine Learning: Projection onto perpendicular spaces is key in algorithms such as PCA and regression, where orthogonal components capture independent variance or noise.\n- Signal Processing: Orthogonal projections isolate meaningful signal components from noise or constraints defined by directional vector spaces.\n- Least Squares: The formula connects directly to the least squares solution, emphasizing the role of inner products in optimal residual minimization.", "---", "### Conclusion", "From the non-negativity of (|\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}|^{\sn]a|, we establish the existence and uniqueness of the projection of (\mathbf{u}) onto the plane perpendicular to (\mathbf{v}). This projection is not only mathematically sound but also essential in decomposing vectors into meaningful geometric components, enabling efficient computation and interpretation across science and engineering.", "---", "Keywords: vector projection, perpendicular projection, inner product, orthogonal decomposition, projection onto plane, (|\mathbf{u} - \mathbf{u} \cdot \mathbf{v} \mathbf{v}| \geq 0), linear algebra, least squares, machine learning, geometric projection."]
