Vectors in Space
Let a,b,c be three noncoplanar vectors. Every vector u in three-dimensional Euclidean space has a
unique representation
Thus the three vectors form a basis of ℝ3.
Figure 9, modernized: vector decomposition in a three-vector spatial basis.
For four noncoplanar points A,B,C,D, every point P can be represented uniquely in affine form
as
Source problem
Through a point P draw lines to the vertices A,B,C,D of a tetrahedron, meeting the opposite face
planes at K,L,M,N. Using directed ratios, prove that the sum of the four ratios in which
K,L,M,N divide PA,PB,PC,PD is −1.
Modern notation references
The notation and terminology in this modernized article follow standard present-day mechanics
and vector-analysis usage, particularly:
- J. R. Taylor, Classical Mechanics, University Science Books, 2005.
- D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge
University Press, 2014.
- H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley,
2002.
Source
This article is a modernized restatement of the corresponding Public Domain article in Louis
Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.”
The original 1930 edition is the source basis.
