Scalar triple product
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The triple scalar product as a "signed" volume
Equations are omitted for technical reasons - download the original pdf
The value [Equation goes here - download the original to see it.] represents a "signed" volume. This means that it can be a positive or a negative quantity. Of course, we are aware that volumes do not strictly take a "sign". However, the sign that this quantity takes is a by-product of the way in which we calculate the volume using vectors. In practical examples it is often just the physical volume of the parallelepiped that is desired, so you simply "drop" the sign at the end of the process. However, the "abstract" meaning of the sign is that if the volume is positive then the vectors making up its edges form a "left-handed set", and if it is negative then they form a "right-handed set". The meaning of "left-handed" and "right-handed" sets are defined as follows. If you make your index, second finger and thumb point at right angles to each other, then you have a left-handed set of perpendicular vectors if you are using your left hand, and a right-handed set of perpendicular vectors if you are using your right hand! [Diagram goes here - download the original to see it.] The scalar-triple product gives an algebraic method of determining whether you have a left or right-hand set [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Evaluating a triple scalar product. Example [Equation goes here - download the original to see it.] Solution [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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