Wednesday, July 25, 2012

Math-Intro to Vectors and Calculus

Introduction to vectors-
A vector is simply an object that has a magnitude and a length in euclidean geometry space.
The components of the vector are defined by y = r sin (θ), and x = r sin (θ) the where theta is the angle between the axis and the vector, and r is defined as the magnitude/length of the vector. If they started out written as components in x and y, you would just see them as (x , y), sometimes the parentheses are substituted with another notation to denote vectors, or x and y will be underlined with a squiggle.
In order to add or subtract two vectors from one another, separate both vectors into their components. Take for example, a vector of length two 30 degrees above the axis, and a vector of length 2 1/2 45 degrees above the axis. First break the two into components thus the first vector becomes (2 (3) 1/2 , 1) and the second vector becomes (1 , 1). Thus the combined result is (2 (3) 1/2 + 1, 2) approximately (4.4, 2).
There are multiple ways to multiply vectors together. The first of which is the most simple, which is to multiply by a scalar. A scalar is a quantity that does not have a direction, and so does not change under coordinate transformations. In this case, you multiply the magnitude of the vector by the scalar, so the r component gets multiplied. For example if you have a vector (1, 1) and you multiply the vector by 2, the magnitude of the vector defined by the pythagorean theorem, (1 + 1 1/2 = (2) 1/2 would become 2 (2) 1/2 , and the components would be (2, 2).
The next way to multiply vectors is called the dot product. A . B which is defined by Σ . This is a scalar multiplication which removes the direction from the vectors and turns it into a scalar. For example, the vector in three space, (1, 2, 3) . (4, 5, 6) would become 1(4) + 2 (5) + 3 (6) = 22. This is equivalent to the magnitude of A multiplied by the magnitude of B multiplied with the cosine of the angle between them.
Another way to multiply vectors together is called the cross product. A x B is equal to magnitude of A multiplied by the magnitude of B sin (θ) in the direction n where n is the normal vector to both A and B. Normal is another term for perpendicular. A more detailed description of the cross product will be given after the introduction to matrices.
Introduction to basic integrals and derivatives-
In order to remember the basic positional equations just use integrals and derivatives to transform from one to another. First we start with some constant acceleration a.
a = a
Then we can integrate this equation with respect to time. Since a is a constant, we can think of this as a x . We then raise 0 to 1 and divide by 1. This gives us the velocity equation.
v = at + v 
The reason why there is a v in this equation is because of the vagueness left by taking a derivative. The derivative of a constant is 0 so a = a + 0, which turns into the constant v0.
This process gets repeated for position.
x = 1/2 a t + v t + x 0
You can also start form the position equation and take derivatives in order to get the other equations of motion. Starting with
x = 1/2 a t + v t + x 
We then take the derivative, multiplying by the exponent then subtracting one from the exponent. giving us
v = at + v 
and so on.

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