Scalar triple product
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Intersecting and parallel lines
If two lines intersect then the shortest distance between them is zero. That means [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] This provides a way of checking whether two lines meet. Example Show that the following two lines [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.] This simplifies the process of finding the shortest distance between two skew lines. However, if the lines are parallel, then the formula breaks down, for we have a different reason why it evaluates to zero. This is because, when d and e are parallel [Equation goes here - download the original to see it.] , so we cannot find a perpendicular vector by this method. So, strictly speaking, in the above example, we should comment firstly that the vectors [Equation goes here - download the original to see it.] are not parallel. (If they would be parallel, one would be the scalar multiple of the other.) But when two lines are parallel, we have to revert to other methods to find the (constant) distance between them; as in the following example. Example [Equation goes here - download the original to see it.] Solution [Diagram goes here - download the original to see it.] Let d be a vector parallel to the lines; for instance, here [Equation goes here - download the original to see it.] a is the position vector of a point A on the line l1 and b is the position vector of a point B on the second line l2. s is a vector perpendicular to both lines. Then [Equation goes here - download the original to see it.] We solve these two equations simultaneously to obtain an expression for the vector s. The length of this vector will be the shortest distance. Thus, let [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] which, on uncoupling, gives [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] which, on uncoupling, gives [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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