Vectors in a Plane

Let $\mathbf{a}$ and $\mathbf{b}$ be nonparallel vectors in a plane. Every vector $\mathbf{u}$ in that plane can be written uniquely as

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

Thus $\{\mathbf{a},\mathbf{b}\}$ is a basis for the plane.

Figure 8, modernized: decomposition of a planar vector in a two-vector basis.

Uniqueness follows from linear independence: if

$\displaystyle \alpha\mathbf{a}+\beta\mathbf{b}=\mathbf0,$

then $\alpha=\beta=0$ .

For noncollinear points $A,B,C$ , every point $P$ in their plane has a unique affine representation

$\displaystyle \boxed{\mathbf r_P= \alpha\mathbf{r}_A+\beta\mathbf{r}_B+\gamma\mathbf r_C, \qquad \alpha+\beta+\gamma=1.} \tag{2}$

These are affine, or barycentric, coordinates relative to the triangle.

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.

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