CSE 5280 Math Fundamentals

Math Fundamentals

- Matrix Math Basics
- Plane and Solid Analytic Geometry,
*Joseph H. Kindle* - WWW Math Tutorial
##### Rectangular Coordinates

Plane is divided into 4 quadrants by 2 perpendicular lines intersecting at the origin point O. The distance from the X axis is the y-coordinate (ordinate) the distance from the Y axis is the x-coordinate (abscissa).

##### Distance between 2 points

The distance between two points P1(x1,y1) and P2(x2,y2) is found by using the Pythagorean Theorem:

D = SQRT( (x2-x1)^2 + (y2-y1)^2)##### Midpoint of a Line

The midpoint, M, of a line segment P1(x1,y1) to P2(x2,y2) is given by:

MP = (x,y) = ((x1+x2/2), (y1+y2)/2)

Slope of a line passing through 2 PointsThe slope of a line is defined as the tangent of the angle of inclination. Thus

*M = TAN(THETA)*where*THETA*is the angle of inclination and*M*is the slope. The slope through 2 points is:

*M = TAN(ø) = (y2 - y1) / (x2 - x1)*

Test for Perpendicular and Parallel LinesTwo lines with slopes

*M1*and*M2*are perpendicular if:*M1 = -1/M2 or M1 * M2 = -1*- the cosine of the angle between them is 0

Two lines are parallel when their slopes M1 and M2 are equal.

Parametric form of a LineGiven points

*P1 = (x1,y1)*and*P2 = (x2,y2)*, the parametric form for a line is:

*x = x1 + t(x2-x1)*is called the

y = y1 + t(y2-y1)

0 <= t <= 1

t*parameter*. When*t*= 0 we get P1 and when*t*= 1 we get P2. As*t*varies between 0 and 1, we get all the other points on the line

segment between P1 and P2.##### Polar Coordinates

A polar coordinate (in 2 dimensions) is an ordered pair (

*r, theta*) which are defined as follows. For a point,*P, r*is the distance from the origin to*P.*Theta is the angle between the x-axis and the line segment from*P*to the origin. Notice that there is more than one valid value for*theta*. For example, if the angle between the x-axis and the line segment is 30 degrees, valid values of*theta*are (in degrees):

30, (30+360) = 390, (390+360) = 750, etc.

The use of polar (instead of Cartesian) coordinates can simplify some calculations.###### Changing Coordinate Systems

Changing from Polar to cartesian coordinates :

*x = r × cos(ø), y = r × sin(ø)*

Conversely, changing from Cartesian coordiantes to Polar:

*r² = x²+ y², ø = arctan(y/x)*

## Vectors

A vector is a directed line segment, characterized by its length and direction.

## Vector Properties

Vectors have special arithmetic properties:

- If
Ais a vector, then-Ais a vector with same length asAbut pointing in the opposite direction.- If A is a vector, the
kAis a vector whose direction is the same as or opposite that of A, depending on the sign of the numberk, and whose length isktimes the length ofA.This is an example of scalar multiplication.- Two vectors can be added together to produce a third vector by using the
parallelogram methodor thehead-to-tail method. This is an example of vector addition.In the parallelogram method, vectors

AandBare placed tail to tail. Their sumA + Bis the vector determined by the diagonal of the parallelogram formed by the vectorsAandB. In thehead-to-tailmethod, the tail ofBis placed at the head ofA. The vectorA + Bis determined by the line segment pointing from the tail ofAto the head ofB.Both methods yield the same results, however thehead-to-tailmethod is easier when adding multiple vectors together. See figure below.

## The dot product

The dot product can be used to generate vectors whose length is 1, this is called

normalizing a vector.To nomalize a vectorv, we simply computev´ = v / ||v||.The resulting vector has a length 1, and is called theunit vector.

Dot products can also be used to measure angles. The angle between the vectorsvanwis

COS - ¹ ( v . w / ||v|| ||w|| )Note that, if

vandware unit vectors, then the division is unnecessary. Dot product characteristics include the following:

- Vectors a and b are perpendicular if (a . b) = 0.
- Vectors a and b are < 90 degrees apart if (a . b) > 0;
- Vectors a and b are > 90 degress apart if (a . b) < 0;
## Example:

Calculate the angle between the vectors

a = 3i + 5jandb = 2i + j,shown the figure below:|| a || = (x_{1})² + (y_{1})² || b || = (x_{2})² + (y_{2})² = (3)² + (5)² = (2)² + (1)² = 9 + 25 = 4 + 1 = SQRT(34) = SQRT(5) a . b = (x_{1}× x_{2}) + (y_{1}× y_{2}) = (3 × 2) + (5 × 1) = 6 + 5 = 11 ø = COS-¹ (11/( SQRT(34) × SQRT(5) )) ø = COS-¹ (11/ SQRT(170)) ø = COS-¹ (11/13.038) = COS-¹(.84368) = 32.47