0. Notation

A triple of numbers $(x,y,z)$ can be viewed as a point in three-dimensional space, in which case we'll usually write $P = (x,y,z)$. It can also be viewed as a vector x or v, in which case we'll often write x$= (x,y,z)$. We may also write x$= (x_1, x_2, x_3)$, especially when talking about computer algorithms. Another alternative is x$= \left [\begin{array}{ccc} x&y&z \end{array} \right ]$. We'll emphasize row vectors, as most computer graphics texts do, rather than column vectors, as many linear algebra texts do. Therefore generally we consider a column vector as the transpose of a row vector and write x$^ t = \left [\begin{array}{c} x\\ y\\ z \end{array} \right ]$. Vectors may be drawn starting from any point, rather than just from the origin.

R$^3 =$ the set of all triples = ``real 3-space''. We usually picture R$^3$ in the way you are used to, with the $x,y$-plane horizontal and the $z$-axis vertical. Sometimes in computer graphics it is better to use other positions, though.

Notation for R$^2$ and R$^n$ is similar. Usually R$^3$ will be used in examples when the dimension doesn't matter.

In R$^3$, the vectors i$,$   j$,$   k are $(1,0,0), (0,1,0), (0,0,1)$, as usual. In R$^2$, i$= (1,0)$ and j$= (0,1)$.

Notice that if x$= (x,y,z)$, then x$= x$   i$+ y$   j$+ z$   k. For this reason, the entries $x,y,z$ can be called the components of x, or the coordinates of x, or the coefficients of x. This equation also illustrates how every vector is a linear combination of i$,$   j$,$k (uniquely). Thus i$,$   j$,$   k are a basis for the vector space R$^3$, called the standard basis. In R$^n$ for general $n$, we write the standard basis as e$^ {(1)},\dots,$   e$^ {(n)}$, so that in R$^3$ we have e$^ {(1)} =$   i, e$^ {(2)} =$   j, e$^ {(3)} =$   k.