Vectors in Space

Let $\mathbf{a},\mathbf{b},\mathbf{c}$ be three noncoplanar vectors. Every vector $\mathbf{u}$ in three-dimensional Euclidean space has a unique representation

$\displaystyle \boxed{\mathbf{u}=\alpha\mathbf{a}+\beta\mathbf{b}+\gamma\mathbf{c}.} \tag{1}$

Thus the three vectors form a basis of $\mathbb{R}^3$ .

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

$\displaystyle \mathbf r_P= \alpha\mathbf{r}_A+\beta\mathbf{r}_B+\gamma\mathbf r_C+ \delta\mathbf r_D, \qquad \alpha+\beta+\gamma+\delta=1.$

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

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.

"vectors in space" is owned by bloftin.
(view preamble)