“parametric equations(of a line in space)”的概念、定义、翻译、参考文献-科学参考

parametric equations(of a line in space)

Mathematics

Given a line in 3-dimensional space, let (x₁,y₁,z₁) be coordinates of a point on the line, and l, m, n be direction ratios of a direction along the line. Then the line consists of all points P whose coordinates (x,y,z) are given by

$x = x_{1} + t l, y = y_{1} + t m, z = z_{1} + t n,$

for some value of the parameter t. These are parametric equations for the line. They are most easily established by using the vector equation of the line and taking components. If none of l, m, n is zero, the equations can be written

$\frac{x - x_{1}}{l} = \frac{y - y_{1}}{m} = \frac{z - z_{1}}{n} (= t),$

which can be considered to be another form of the parametric equations, or called the equations of the line in ‘symmetric form’. If, say, n = 0 and l and m are both non-zero, the equations are written

$\frac{x - x_{1}}{l} = \frac{y - y_{1}}{m}, z = z_{1} .$

if, say, m = n = 0, they become y = y₁, z = z₁.

More generally in n-dimensional space, if p and a are in ℝⁿ with a ≠ 0, then

$r (t) = p + t a (t \in ℝ)$

is a parameterization of the line passing through p which is parallel to a.

单词	parametric equations(of a line in space)
释义	parametric equations(of a line in space) Mathematics Given a line in 3-dimensional space, let (x₁,y₁,z₁) be coordinates of a point on the line, and l, m, n be direction ratios of a direction along the line. Then the line consists of all points P whose coordinates (x,y,z) are given by $x = x_{1} + t l, y = y_{1} + t m, z = z_{1} + t n,$ for some value of the parameter t. These are parametric equations for the line. They are most easily established by using the vector equation of the line and taking components. If none of l, m, n is zero, the equations can be written $\frac{x - x_{1}}{l} = \frac{y - y_{1}}{m} = \frac{z - z_{1}}{n} (= t),$ which can be considered to be another form of the parametric equations, or called the equations of the line in ‘symmetric form’. If, say, n = 0 and l and m are both non-zero, the equations are written $\frac{x - x_{1}}{l} = \frac{y - y_{1}}{m}, z = z_{1} .$ if, say, m = n = 0, they become y = y₁, z = z₁. More generally in n-dimensional space, if p and a are in ℝⁿ with a ≠ 0, then $r (t) = p + t a (t \in ℝ)$ is a parameterization of the line passing through p which is parallel to a.
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