1. CSC 123 – Computational Art Points and Vectors
Zoë Wood

2. Representing Points and Vectors
A 2D point:
Represents a location with respect to some coordinate system
A 2D vector:
Represents a displacement from a position

3. Vectors Objects with length and direction.
Represent motion between points in space.
We will think of a vector as a displacement from one point to another.
We use bold typeface to indicate vectors. v

4. Vectors as Displacements
As a displacement, a vector has length and direction, but no inherent location.
It doesn’t matter if you step one foot forward in Boston or in New York. v v v v v v

5. Vector Notation We represent vectors as a list of components.
By convention, we write vectors as a column matrix:
We can also write them as row matrix, indicated by vT:
vT = [x0, x1] or wT = [x0, x1 , x2]
All components of the zero vector 0 are 0. or

6. Vector Operations Addition and multiplication by a scalar: Examples:
a + b
s*a
Examples:

7. Vector Operations Addition and multiplication by a scalar: Examples:
a + b
s*a
Examples:

8. Vector Scaling – geometric interpretation
Multiplying a vector by a scalar multiplies the vector’s length without changing its direction:
3.5 * v
2 * v v

9. Vector scalar Multiplication
Given wT = [x0, x1 , x2] and scalars s and t
s * wT = [s * x0, s * x1 , s * x2]
Properties of Vector multiplication
Associative: (st)V = s(tV)
Multiplicative Identity: 1V = V
Scalar Distribution: (s+t)V = sV+tV
Vector Distribution: s(V+W) = sV+sW

10. Vector addition – geometric interpretation
Paralleologram rule
To visualize what vector addition is doing, here is a 2D example:
V+W V W

11. Vector Addition Head to tail: At the end of a, place the start of b.
a + b = b + a
Components: a + b = [a0 + b0, a1 + b1]T
a+b b a

12. Unary Minus Algebraically, -v = (-1) v.
The geometric interpretation of -v is a vector in the opposite direction: v -v

13. Vector Subtraction b - a = -a + b
Components: b - a = [b0 - a0, b1 - a1]T
-a+b b a
b-a

14. Linear Combinations A linear combination of m vectors
is a vector of the form:
where a1, a2, …, am are scalars.
Example:

15. Linear Combinations A linear combination of m vectors
is a vector of the form:
where a1, a2, …, am are scalars.
Example:

16. Vector Addition Given vT = [x0, x1 , x2] and wT = [y0, y1 , y2]
V+W = [x0 +y0, x1 + y1 , x2 + y2]
Properties of Vector addition
Commutative: V+W=W+V
Associative (U+V)+W = U+(V+W)
Additive Identity: V+0 = V
Additive Inverse: V+W = 0, W=-V

17. Points and Vectors We normally say v = [3 2 7]T.
Similarly, for a point we say P = [3 2 7]T.
But points and vectors are very different:
Points: have a location, but no direction.
Vectors: have direction, but no location.
Point: A fixed location in a geometric world.
Vector: Displacement between points in the world.

18. Vectors to Specify Locations
Locations can be represented as displacements from another location.
There is some understood origin location O from which all other locations are stored as offsets.
Note that locations are not vectors!
Locations are described using points. O v P

19. Operations on Vectors and Points
Linear operations on vectors make sense, but the same operations do not make sense for points:
Addition:
Vectors: concatenation of displacements.
Points: ?? (SLO + LA = ?)
Multiplication by a scalar:
Vectors: increases displacement by a factor.
Points: ?? (What is 7 times the North Pole?)
Zero element:
Vectors: The zero vector represents no displacement.
Points: ?? (Is there a zero point?)

20. Operations on Points However, some operations with points make sense.
We can subtract two points. The result is the motion from one point to the other:
P - Q = v
We can also start with a point and move to another point:
Q + v = P

21. Other operations There are other operations on 3D vectors
dot product
cross product
We won’t talk about these

22. Vector Basis Vectors which are linearly independent form a basis.
The vectors are called basis vectors.
Any vector can be expressed as linear combination of the basis vectors:
Note that the coefficients ac and bc are unique.

23. Orthogonal Basis
If the two vectors are at right angles to each other, we call the it an orthogonal basis.
If they are also vectors with length one, we call it an orthonormal basis.
In 2D, we can use two such special vectors x and y to define the Cartesian basis vectors
x = [1 0] y = [0 1] y x

24. By Pythagorean theorem:
Cartisian
Descarté
One reason why cartesian coordinates are nice
By Pythagorean theorem:
length of c is: y c
0.5 2 x

25. Normalizing a vector
In order to make a vector be a ‘normal’ vector it must be of unit length.
We can normalize a vector by dividing each component by its length

26. How can we use these?
When we want to control movement, we can use vectors
For example if we want to have a faces’ eyeballs follow the mouse…

27. Problem Move eyeballs to the edge of a circle in a given direction
Compute the vector, which is a displacement from the current position to where the mouse was clicked
Scale the vector to only be as long as the eye’s radius
Displace the eyeball to rim of the eye

28. The Basis Matters
When we describe a vector, we need to agree on which basis we use. v -2 y 5 x

29. The Basis Matters
Here is the same vector, expressed using a different vector basis. a b v

30. Linear Independence
A 2D vector can be written as a linear combination of any two non-zero vectors which are not parallel.
This property of the two vectors is called linear independence (i.e., not parallel) b
c = 2a + 0.5b
0.5b c 2a a