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  "body": "\n## Definition\n\nLet $I \\subseteq \\mathbb{R}$ be an interval, let $\\gamma \\colon I \\to \\mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]], and let $t_0 \\in I$.\nIf the limit\n$$\\gamma'(t_0) = \\lim_{h \\to 0} \\frac{\\gamma(t_0 + h) - \\gamma(t_0)}{h} = \\big( \\gamma_1'(t_0), \\dots, \\gamma_n'(t_0) \\big)$$\nexists, it is the tangent vector, or velocity vector, of $\\gamma$ at $t_0$.\nIts norm $\\|\\gamma'(t_0)\\|$ is the speed of $\\gamma$ at $t_0$.\nIf $\\gamma'(t_0) \\ne 0$, the unit tangent vector is $T(t_0) = \\gamma'(t_0) / \\|\\gamma'(t_0)\\|$, and the tangent line at $\\gamma(t_0)$ is $\\{\\gamma(t_0) + s \\, \\gamma'(t_0) : s \\in \\mathbb{R}\\}$.\n\n## Remarks\n\nAt an end point of $I$, the limit is one-sided.\nFor the circle $\\gamma(t) = (\\cos t, \\sin t)$, $\\gamma'(t) = (-\\sin t, \\cos t)$. It is orthogonal to $\\gamma(t)$ and has length $1$.\nThe tangent vector can be zero. For $\\gamma(t) = (t^2, t^3)$, $\\gamma'(0) = (0, 0)$, and the image has a cusp at the origin.\nThe tangent vector depends on the parametrisation, but for a [[regular curve|06JMR2XWXW]] its direction does not change under an increasing change of parameter.\n\n## Explanations\n\n### L1 human\n\nThe tangent vector points in the direction in which the curve moves at time $t_0$.\nIts length tells how fast the point moves.\n",
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