Vectors - Mathematics for Computer Graphics - page 44

Let's now consider two vectors r and s and compute the normal vector t .
The vectors will be chosen so that we can anticipate approximately the answer.
Figure 6.10 shows the vectors r and s and the normal vector t . Table 6.2
contains the coordinates of the vertices forming the two vectors.

⎡ ⎤ ⎡ ⎤

x 3 − x 2 x 1 − x 2

⎣ ⎦ s = ⎣ ⎦

r = y 3 − y 2 y 1 − y 2

z 3 − z 2 z 1 − z 2

r = − i + j

s = − i + k

r × s =(1 × 1 − 0 × 0) i − ( − 1 × 1 − ( − 1) × 0) j

+( − 1 × 0 − ( − 1) × 1) k

= i + j + k

This confirms what we expected from Figure 6.10. Let's now reverse the vec-
tors to illustrate the importance of vector sequence:

s = − i + k

r = − i + j

Y

n 3

r

t

s

n 1 n 2

Z X

Fig. 6.10. The vector t is normal to the vectors r and s .

Table 6.2. Coordinates of the vertices used
in Fig. 6.10.

Vertex x y z

v 1 0 0 1

v 2 1 0 0

v 3 0 1 0