[<< wikibooks] Linear Algebra/Projection
This section is optional; only the last two sections of Chapter Five require this material.
We have described the projection

π

{\displaystyle \pi }
from

R

3

{\displaystyle \mathbb {R} ^{3}}
into its

x
y

{\displaystyle xy}
plane subspace as a "shadow map".
This shows why, but it also shows that some shadows fall upward.

So perhaps a better description is: the projection of

v
→

{\displaystyle {\vec {v}}}

is the

p
→

{\displaystyle {\vec {p}}}
in the plane with the property that
someone standing on

p
→

{\displaystyle {\vec {p}}}
and looking straight up or down sees

v
→

{\displaystyle {\vec {v}}}
.
In this section we will generalize this to other projections,
both orthogonal (i.e., "straight up and down") and nonorthogonal.