Scalar triple product
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Shortest distance between two skew lines
The shortest distance between two lines is the length of the line that is perpendicular to both lines. [Diagram goes here - download the original to see it.] The vector equation of the line [Equation goes here - download the original to see it.], where d is a vector parallel to the line, and a is the position vector of a point A on the line. Likewise, the vector equation of the line[Equation goes here - download the original to see it.], where e is a vector parallel to the line, and b is the position vector of a point on the line. [Diagram goes here - download the original to see it.] The vector [Equation goes here - download the original to see it.] is perpendicular to both d and e. A unit vector, therefore, parallel to the shortest line between the two lines is give by [Equation goes here - download the original to see it.] Now the vector [Equation goes here - download the original to see it.] is the vector joining B to A. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] This may have a "sign" (positive or negative sign) depending on whether [Equation goes here - download the original to see it.]forms a left or right-hand set. As we are only interested in the magnitude of the shortest distance, we take the modulus of the above expression [Equation goes here - download the original to see it.] This expression contains a triple scalar product. Using this formula we can find the shortest distance between two skew lines. (Lines are skew when they form a non-zero angle between their directions, and when they do not intersect. Example [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.]
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Contents of
Scalar triple product
1 Scalar Triple Product
2 The triple scalar product as a "signed" volume
3 Shortest distance between two skew lines
4 Intersecting and parallel lines
5 Co-planar points
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