1. CS 3388 Working with 3D Vectors
[Hill §4.1—4.4] x z y
(1,2,0)
(1,2,2)
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAAAA

2. Vector Spaces
The space Rn is a vector space because it is closed under
Setting n=3 is special case
vector addition:
u,v 2 Rn ) u + v 2 Rn
scalar multiplication:
v 2 Rn, ® 2 R ) ®v 2 Rn

3. k-Flats and Subspaces A subspace is a subset of points
A k-flat is a subspace “congruent” to a vector space of dimension k
A point is a 0-flat
A line is a 1-flat
A plane is a 2-flat y u v z x u

4. Vector Norm
Identities:
(triangle inequality)

5. 3D Dot Product Also known as inner- or scalar product since Notice
If assume (column vectors)

6. Dot Product
Identities:
Geometry: x z y b a
Therefore…

7. Dot Product as Projection
The value is the length of a projected onto b multiplied by
If then a b a b

8. Sign of Dot Product If then a and b facing similar directions
If then a and b facing opposite directions
Useful for detecting front-facing and back-facing surfaces (much later a

9. Cross Product Also called vector product, as Way to remember order:
(2£2 determinants)
careful!

10. Geometry of Cross Product
The vector where
If and
then
If then
“c has unit norm”
“c orthogonal to a and to b” x z y b a c

11. Cross Product Identities
these follow from

12. Normal Vectors Let be a surface (dimension 2)
For each surface point define
A normal at is any
such that for all n v p q p

13. Normals for Planes For a plane (2-flat), point doesn’t matter because
Given with can compute a normal
Can calculate u,v from non-colinear points x z y n u v
(vectors tangential to plane do not change with position) y q p r z x

14. (i.e. do it on blackboard)
Calculating a Normal
Let define a plane y q p r z x
“Calculemus!”
---Leibniz
(i.e. do it on blackboard) y n z x