“distance between two lines(in 3‐dimensional space)”的概念、定义、翻译、参考文献-科学参考

distance between two lines(in 3‐dimensional space)

Mathematics

Let l₁ and l₂ be lines in space that do not intersect. There are two cases. If l₁ and l₂ are parallel, the distance between the two lines is the length of any line segment N₁ N₂, with N₁ on l₁ and N₂ on l₂ perpendicular to both lines. If l₁ and l₂ are not parallel, there are unique points N₁ on l₁ and N₂ on l₂ such that the length of the line segment N₁N₂ is the shortest possible. The length |N₁N₂| is the distance between the two lines. In fact, the line N₁N₂ is the common perpendicular of l₁ and l₂.

If the two lines are skew and have equations r × a₁= b₁ and r × a₂ = b₂ where × denotes the vector product, then the distance between the lines equals

$\frac{| a_{1} ∙ b_{2} + a_{2} ∙ b_{1} |}{| a_{1} \times a_{2} |} .$

单词	distance between two lines(in 3‐dimensional space)
释义	distance between two lines(in 3‐dimensional space) Mathematics Let l₁ and l₂ be lines in space that do not intersect. There are two cases. If l₁ and l₂ are parallel, the distance between the two lines is the length of any line segment N₁ N₂, with N₁ on l₁ and N₂ on l₂ perpendicular to both lines. If l₁ and l₂ are not parallel, there are unique points N₁ on l₁ and N₂ on l₂ such that the length of the line segment N₁N₂ is the shortest possible. The length \|N₁N₂\| is the distance between the two lines. In fact, the line N₁N₂ is the common perpendicular of l₁ and l₂. If the two lines are skew and have equations r × a₁= b₁ and r × a₂ = b₂ where × denotes the vector product, then the distance between the lines equals $\frac{\| a_{1} ∙ b_{2} + a_{2} ∙ b_{1} \|}{\| a_{1} \times a_{2} \|} .$
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