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  "body": "\n## Description\n\nThe Euclidean plane is the set $\\mathbb{R}^2$ of the ordered pairs $(x, y)$ of real numbers, with the dot product $(x_1, y_1) \\cdot (x_2, y_2) = x_1 x_2 + y_1 y_2$.\nThe elements of $\\mathbb{R}^2$ are the points of the plane.\nThe dot product gives the length of a vector, the distance between two points and the angle between two vectors.\nThe distance between the points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ is $\\sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2}$.\nThe Euclidean plane is the [[Euclidean space|0XPR1E85W0]] of dimension 2.\n\n## Remarks\n\nEach Euclidean space of dimension 2 is isometric to $\\mathbb{R}^2$. A choice of a [[Cartesian coordinate system|0RWKYX8339]] gives such an isometry.\nThe plane with a fixed Cartesian coordinate system is also called the Cartesian plane.\nBooks I to IV and VI of Euclid's Elements study the geometry of the plane.\n\n## Explanations\n\n### L1 human\n\nThink of the Euclidean plane as an infinite flat sheet of paper. Two numbers, the coordinates $x$ and $y$, fix the position of each point. For example, the points $(0, 0)$ and $(3, 4)$ are at distance $\\sqrt{9 + 16} = 5$.\n",
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