{
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  "body": "\n## Summary\n\nA parametric equation of a [[line|0MP9BCF6DP]] describes its points as the values of a function of one real parameter $t$.\nThe line in $\\mathbb{R}^3$ through the point $(x_0, y_0, z_0)$ with the nonzero direction vector $(a, b, c)$ has the parametric equations\n$$x = x_0 + a t, \\quad y = y_0 + b t, \\quad z = z_0 + c t, \\qquad t \\in \\mathbb{R}.$$\nIn vector form, the line through the point $p$ with the direction vector $v \\neq 0$ is the set of the points $p + t v$ for $t \\in \\mathbb{R}$. The same form works in $\\mathbb{R}^n$ for each $n \\geq 2$.\n\n## Remarks\n\nThe line through two distinct points $A$ and $B$ has the parametric equation $A + t(B - A)$. The values $0 \\leq t \\leq 1$ give the segment from $A$ to $B$, and $t = 1/2$ gives its midpoint.\nIf $a$, $b$ and $c$ are all nonzero, you can remove $t$. The result is the symmetric form $\\frac{x - x_0}{a} = \\frac{y - y_0}{b} = \\frac{z - z_0}{c}$.\n\n## Explanations\n\n### L1 human\n\nThink of $t$ as time and of $p + t v$ as the position of a particle. At time $0$ the particle is at $p$, and it moves with the constant velocity $v$. Its path is the line.\n",
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  "msc": [
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