Find the minimum distance between the point (1,0,0)^T and...

Find the minimum distance between the point $(1,0,0)^T$ and the plane $x+y-z=0$ when distance is measured in (a) the Euclidean norm; (b) the weighted norm $\|\mathbf{w}\|=$ $\sqrt{w_1^2+2 w_2^2+3 w_3^2} ;$ (c) the norm based on the positive definite matrix $\left(\begin{array}{rrr}3 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 3\end{array}\right)$.

Key Concepts

Projection onto a Plane in Different Norms

Projection onto a plane in the context of different norms involves finding the point on the plane that minimizes the distance to a given point with respect to the specified norm. While the Euclidean case uses orthogonal projection based on the standard inner product, the weighted and matrix-induced norms require modifying the projection process—often via Lagrange multipliers or solving a constrained optimization problem—to account for the altered geometry. This concept is fundamental in optimization and geometry when dealing with distance minimization problems under various metrics.

Euclidean Norm

The Euclidean norm is the standard way of measuring distance in space, defined as the square root of the sum of the squares of the coordinates. In problems involving the distance from a point to a plane, it leads to the familiar formula that relies on the plane’s normal vector. This concept is central in geometry and analysis as it provides the basis for measuring lengths and angles in Euclidean space.

Weighted Norm

A weighted norm is a modification of the Euclidean norm where different coordinates are scaled by different weights, resulting in an ellipsoidal distance measure rather than a spherical one. This type of norm is defined by taking a weighted sum of squares of the components, which effectively alters the geometry of the space, and is useful when different dimensions have different levels of importance or units.

Matrix-Induced Norm (Quadratic Form Norm)

A matrix-induced norm is defined via a positive definite matrix and takes the form of a quadratic expression. Specifically, the norm of a vector is given by the square root of its transpose multiplied by the positive definite matrix and then by the vector. This generalizes the Euclidean norm and is closely related to the Mahalanobis distance; it is particularly significant in optimization and statistics where anisotropic scaling and correlations between variables are present.

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