📂Geometry

General Definitions of Lines, Planes, and Spheres

Definition ¹

Let’s assume that a vector space $X$ is given.

1. The set of points satisfying the following equation $L \subset X$ or $\alpha (t)$ itself is defined as a Line that passes through point $\mathbf{x}_{0} \in X$ and is parallel to vector $\mathbf{v} \ne 0$. $$\alpha (t) = \mathbf{x}_{0} + t \mathbf{v} \qquad , t \in \mathbb{R}$$
2. The set of points satisfying the following equation $P \subset X$ is defined as a Plane that passes through point $\mathbf{x}_{0} \in X$ and is perpendicular to vector $\mathbf{n} \ne 0$. $$\left< \mathbf{x} - \mathbf{x}_{0} , \mathbf{n} \right> = \mathbf{0}$$
3. The set of points satisfying the following equation $S \subset X$ is defined as a Sphere with center $\mathbf{x}_{0} \in X$ and Radius $r > 0$. $$\left< \mathbf{x} - \mathbf{x}_{0} , \mathbf{x} - \mathbf{x}_{0} \right> = r^{2}$$

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• $\left< \cdot , \cdot \right>$ is the dot product.

Line, Plane, and Sphere all in one

LOL