1. Chapter 4.1 Mathematical Concepts

2. Applied Trigonometry Trigonometric functions
Defined using right triangle h y a x

3. Applied Trigonometry Angles measured in radians
Full circle contains 2p radians

4. Applied Trigonometry
Sine and cosine used to decompose a point into horizontal and vertical components y r
r sin a a x
r cos a

5. Applied Trigonometry
Trigonometric identities

6. Applied Trigonometry Inverse trigonometric functions
Return angle for which sin, cos, or tan function produces a particular value
If sin a = z, then a = sin-1 z
If cos a = z, then a = cos-1 z
If tan a = z, then a = tan-1 z

7. Applied Trigonometry Law of sines Law of cosines
Reduces to Pythagorean theorem when g = 90 degrees a c b b g a

8. Vectors and Matrices
Scalars represent quantities that can be described fully using one value
Mass
Time
Distance
Vectors describe a magnitude and direction together using multiple values

9. Vectors and Matrices Examples of vectors Difference between two points
Magnitude is the distance between the points
Direction points from one point to the other
Velocity of a projectile
Magnitude is the speed of the projectile
Direction is the direction in which it’s traveling
A force is applied along a direction

10. Vectors and Matrices Vectors can be visualized by an arrow
The length represents the magnitude
The arrowhead indicates the direction
Multiplying a vector by a scalar changes the arrow’s length 2V V –V

11. Vectors and Matrices
Two vectors V and W are added by placing the beginning of W at the end of V
Subtraction reverses the second vector W V
V + W W –W V
V – W V

12. Vectors and Matrices
An n-dimensional vector V is represented by n components
In three dimensions, the components are named x, y, and z
Individual components are expressed using the name as a subscript:

13. Vectors and Matrices
Vectors add and subtract componentwise

14. Vectors and Matrices
The magnitude of an n-dimensional vector V is given by
In three dimensions, this is

15. Vectors and Matrices
A vector having a magnitude of 1 is called a unit vector
Any vector V can be resized to unit length by dividing it by its magnitude:
This process is called normalization

16. Vectors and Matrices
A matrix is a rectangular array of numbers arranged as rows and columns
A matrix having n rows and m columns is an n  m matrix
At the right, M is a 2  3 matrix
If n = m, the matrix is a square matrix

17. Vectors and Matrices
The entry of a matrix M in the i-th row and j-th column is denoted Mij
For example,

18. Vectors and Matrices
The transpose of a matrix M is denoted MT and has its rows and columns exchanged:

19. Vectors and Matrices
An n-dimensional vector V can be thought of as an n  1 column matrix:
Or a 1  n row matrix:

20. Vectors and Matrices Product of two matrices A and B
Number of columns of A must equal number of rows of B
Entries of the product are given by
If A is a n  m matrix, and B is an m  p matrix, then AB is an n  p matrix

21. Vectors and Matrices
Example matrix product

22. Vectors and Matrices
Matrices are used to transform vectors from one coordinate system to another
In three dimensions, the product of a matrix and a column vector looks like:

23. Vectors and Matrices The n  n identity matrix is denoted In
For any n  n matrix M, the product with the identity matrix is M itself
InM = M
MIn = M
The identity matrix is the matrix analog of the number one
In has entries of 1 along the main diagonal and 0 everywhere else

24. Vectors and Matrices
An n  n matrix M is invertible if there exists another matrix G such that
The inverse of M is denoted M-1

25. Vectors and Matrices Not every matrix has an inverse
A noninvertible matrix is called singular
Whether a matrix is invertible can be determined by calculating a scalar quantity called the determinant

26. Vectors and Matrices
The determinant of a square matrix M is denoted det M or |M|
A matrix is invertible if its determinant is not zero
For a 2  2 matrix,

27. The Dot Product
The dot product is a product between two vectors that produces a scalar
The dot product between two n-dimensional vectors V and W is given by
In three dimensions,

28. The Dot Product a is the angle between the two vectors
The dot product satisfies the formula
a is the angle between the two vectors
Dot product is always 0 between perpendicular vectors
If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite

29. The Dot Product
The dot product of a vector with itself produces the squared magnitude
Often, the notation V 2 is used as shorthand for V  V

30. The Dot Product
The dot product can be used to project one vector onto another V a W

31. The Cross Product
The cross product is a product between two vectors the produces a vector
The cross product only applies in three dimensions
The cross product is perpendicular to both vectors being multiplied together
The cross product between two parallel vectors is the zero vector (0, 0, 0)

32. The Cross Product The cross product between V and W is
A helpful tool for remembering this formula is the pseudodeterminant

33. The Cross Product
The cross product can also be expressed as the matrix-vector product
The perpendicularity property means

34. The Cross Product
The cross product satisfies the trigonometric relationship
This is the area of the parallelogram formed by V and W V
||V|| sin a a W

35. The Cross Product
The area A of a triangle with vertices P1, P2, and P3 is thus given by

36. The Cross Product Cross products obey the right hand rule
If first vector points along right thumb, and second vector points along right fingers,
Then cross product points out of right palm
Reversing order of vectors negates the cross product:
Cross product is anticommutative

37. Stop Here

38. Transformations
Calculations are often carried out in many different coordinate systems
We must be able to transform information from one coordinate system to another easily
Matrix multiplication allows us to do this

39. Transformations
Suppose that the coordinate axes in one coordinate system correspond to the directions R, S, and T in another
Then we transform a vector V to the RST system as follows

40. Transformations
We transform back to the original system by inverting the matrix:
Often, the matrix’s inverse is equal to its transpose—such a matrix is called orthogonal

41. Transformations
A 3  3 matrix can reorient the coordinate axes in any way, but it leaves the origin fixed
We must at a translation component D to move the origin:

42. Transformations Homogeneous coordinates Four-dimensional space
Combines 3  3 matrix and translation into one 4  4 matrix

43. Transformations V is now a four-dimensional vector
The w-coordinate of V determines whether V is a point or a direction vector
If w = 0, then V is a direction vector and the fourth column of the transformation matrix has no effect
If w  0, then V is a point and the fourth column of the matrix translates the origin
Normally, w = 1 for points

44. Transformations
The three-dimensional counterpart of a four-dimensional homogeneous vector V is given by
Scaling a homogeneous vector thus has no effect on its actual 3D value

45. Transformations
Transformation matrices are often the result of combining several simple transformations
Translations
Scales
Rotations
Transformations are combined by multiplying their matrices together

46. Transformations Translation matrix
Translates the origin by the vector T

47. Transformations Scale matrix Scales coordinate axes by a, b, and c
If a = b = c, the scale is uniform

48. Transformations Rotation matrix
Rotates points about the z-axis through the angle q

49. Transformations
Similar matrices for rotations about x, y

50. Transformations
Normal vectors transform differently than do ordinary points and directions
A normal vector represents the direction pointing out of a surface
A normal vector is perpendicular to the tangent plane
If a matrix M transforms points from one coordinate system to another, then normal vectors must be transformed by (M-1)T

51. Geometry A line in 3D space is represented by
S is a point on the line, and V is the direction along which the line runs
Any point P on the line corresponds to a value of the parameter t
Two lines are parallel if their direction vectors are parallel

52. Geometry
A plane in 3D space can be defined by a normal direction N and a point P
Other points in the plane satisfy N Q P

53. Geometry A plane equation is commonly written
A, B, and C are the components of the normal direction N, and D is given by
for any point P in the plane

54. Geometry A plane is often represented by the 4D vector (A, B, C, D)
If a 4D homogeneous point P lies in the plane, then (A, B, C, D)  P = 0
If a point does not lie in the plane, then the dot product tells us which side of the plane the point lies on

55. Geometry
Distance d from a point P to a line S + t V P d S V

56. Geometry Use Pythagorean theorem: Taking square root,
If V is unit length, then V 2 = 1

57. Geometry Intersection of a line and a plane
Let P(t) = S + t V be the line
Let L = (N, D) be the plane
We want to find t such that L  P(t) = 0
Careful, S has w-coordinate of 1, and V has w-coordinate of 0

58. Geometry
If L  V = 0, the line is parallel to the plane and no intersection occurs
Otherwise, the point of intersection is