Understanding Geometric Algebra

Rotation and geometric product  87

a

l

x

x

x

x

x

x

x

x

x

n

x

(a) (b)

FIGURE 6.1 (a) The projection x

k

, th e rejection x

⊥

, and the reflection x

⊤

of vector x for a line

in the direction a. (b) The projection x

k

, the rejection x

⊥

, and the reflection x

⊤

of vector x for a

plane with un it surface normal n.

The projection x

k

of vector x onto a plane with unit surface normal n equals its re jection

from the surface normal (Fig. 6.1(b)), so x

k

= (x ∧ n)n

−1

from Eq. (6.58). Conversely,

the rejection x

⊥

from this plane equals its projection onto the surface normal, so x

⊥

=

hx, nin

−1

from Eq. (6.5 8). The reflection x

⊤

of vector x with respect to this plane is

obtained by subtracting ...