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  "body": "\n## Definition\n\nLet $n$ be a positive integer, and let $u$ and $v$ be nonzero [[vectors|0QN3D80AWP]] in $\\mathbb{R}^n$.\nLet $u \\cdot v = u_1 v_1 + \\dots + u_n v_n$ be their dot product, and let $\\|u\\|$ and $\\|v\\|$ be their [[lengths|0XKBKGTCPC]].\nThe angle between $u$ and $v$ is the unique real number $\\theta$ with $0 \\leq \\theta \\leq \\pi$ such that\n$$\\cos \\theta = \\frac{u \\cdot v}{\\|u\\| \\, \\|v\\|}.$$\nThe Cauchy–Schwarz inequality $|u \\cdot v| \\leq \\|u\\| \\, \\|v\\|$ makes sure that the quotient is in $[-1, 1]$. Thus $\\theta = \\arccos\\left(\\frac{u \\cdot v}{\\|u\\| \\, \\|v\\|}\\right)$ exists.\n\n## Remarks\n\nThe angle is $0$ if and only if $v = c u$ for a real number $c \u003e 0$. It is $\\pi$ if and only if $v = c u$ for a real number $c \u003c 0$.\nThe angle does not depend on the order of the vectors. It does not change when you multiply $u$ or $v$ by a positive number.\nMathlib uses the same formula for all vectors. With the convention $0 / 0 = 0$, the angle between the zero vector and any vector is then $\\pi / 2$.\n\n## Explanations\n\n### L1 human\n\nPut the two arrows at the same start point. The angle between them is the opening between the arrows, measured in radians from $0$ to $\\pi$. For example, $(1, 0)$ and $(0, 1)$ have dot product $0$, so the angle between them is $\\arccos 0 = \\pi / 2$, a right angle.\n",
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