“line(in three dimensions)”的概念、定义、翻译、参考文献-科学参考

line(in three dimensions)

Mathematics

As in two dimensions there are a number of standard formats which can be used.
Two‐point form.
The condition that the general point P(x,y,z) lies on the line through A(x₁,y₁,z₁) and B(x₂,y₂,z₂) is equivalent to requiring the direction ratios of AP and AB being equal. That is:

$\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}} .$
Parametric vector form.
If the vector (l,m,n) is parallel to the line and the point A(x₁,y₁,z₁) lies on it, the position vector of a general point P on the line has the form

$r = OP = (\begin{matrix} x_{1} \\ y_{1} \\ z_{1} \end{matrix}) + λ (\begin{matrix} l \\ m \\ n \end{matrix}) .$

If a and b are the position vectors of points A and B on the line, the position vector of a general point is

$r = a + λ (b - a) .$
Vector form.
The equation r = a + λ‎(b−a) can be rewritten as (r−a)×(b−a) = 0, where × denotes the vector product. More generally the equation

$r \times a = b,$

defines a line when a ≠ 0 and a⋅b = 0. This line is parallel to a and passes through the point (a × b)/|a|².

单词	line(in three dimensions)
释义	line(in three dimensions) Mathematics As in two dimensions there are a number of standard formats which can be used. Two‐point form. The condition that the general point P(x,y,z) lies on the line through A(x₁,y₁,z₁) and B(x₂,y₂,z₂) is equivalent to requiring the direction ratios of AP and AB being equal. That is: $\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}} .$ Parametric vector form. If the vector (l,m,n) is parallel to the line and the point A(x₁,y₁,z₁) lies on it, the position vector of a general point P on the line has the form $r = OP = (\begin{matrix} x_{1} \\ y_{1} \\ z_{1} \end{matrix}) + λ (\begin{matrix} l \\ m \\ n \end{matrix}) .$ If a and b are the position vectors of points A and B on the line, the position vector of a general point is $r = a + λ (b - a) .$ Vector form. The equation r = a + λ‎(b−a) can be rewritten as (r−a)×(b−a) = 0, where × denotes the vector product. More generally the equation $r \times a = b,$ defines a line when a ≠ 0 and a⋅b = 0. This line is parallel to a and passes through the point (a × b)/\|a\|².
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Two‐point form.The condition that the general point P(x,y,z) lies on the line through A(x1,y1,z1) and B(x2,y2,z2) is equivalent to requiring the direction ratios of AP and AB being equal. That is: x−x1x2−x1=y−y1y2−y1=z−z1z2−z1.

Vector form.The equation r = a + λ‎(b−a) can be rewritten as (r−a)×(b−a) = 0, where × denotes the vector product. More generally the equation r×a=b, defines a line when a ≠ 0 and a⋅b = 0. This line is parallel to a and passes through the point (a × b)/|a|2.

Vector form.
The equation r = a + λ‎(b−a) can be rewritten as (r−a)×(b−a) = 0, where × denotes the vector product. More generally the equation

$r \times a = b,$

defines a line when a ≠ 0 and a⋅b = 0. This line is parallel to a and passes through the point (a × b)/|a|².