Lecture 5: Vectors and Two Dimensional Motion
Definitions
Scalar Quantity = any quantity with only magnitude and no direction. Examples: mass, time, distance

Vector Quantity = any quantity with both magnitude and direction. Examples: velocity, acceleration, and displacement

Coordinate System or Frame of Reference = a way for us to describe an object's location in space. Most common ones used: Cartesian Coordinates (x, y, z) and polar coordinates  (r, q)

Coordinate Systems:
                    The first thing we need to do when dealing with vector quantities is to establish the coordinate system being used in order to locate objects properly in space. Setting up frames of reference involves picking an origin.  The choice is totally up to you, however, the choice should be one that makes the problem easier to deal with and easier to visualize.
                    To set up Cartesian coordinates: pick an origin and set up the axes (as shown in the figure below) and label the axes.  Each Cartesian coordinate is made up of an x and a y component written as (x,y).

                    Plane polar coordinates are set up on a system much like the Cartesian coordinates, the only difference is that the polar coordinate specifies a radius, r, and the angle that the radius makes with the positive x-axis, q, as shown in the figure below.
Converting from Cartesian to Polar coordinates
r = [x2 + y2]1/2
q = tan-1(y/x)
Converting from Polar to Cartesian:
x = r cos q
y = r sin q

Vectors and their Properties
Vector Equality
                    Two vectors are equal if and only if, they have the same magnitude and they point in the same direction. Vectors are drawn as arrows, the length of the arrow represents the magnitude, and the direction of the arrow establishes the direction of the vector.

Graphical Vector Addition
Method 1
                    The sum of 2 or more vectors can be found graphically using the following steps (consider adding 2 vectors : A and B, illustrated in the figure below:
                        Draw the first vector: A
                        Draw the second vector ,B with its tail placed at the tip of vector A
                        Draw the resultant vector by drawing a vector from the tail of A to the tip of B

Method 2: Parallelogram Method:
                        Draw Vector A and B tail to tail with an appropriate coordinate system.
                        Draw imaginary vectors A and B (represented by the dotted arrows in the figure)
                     The resultant vector C is just the diagonal of the parallelogram formed by vectors A and B and the imaginary vectors A and B.
Other Properties of Vectors
Associative law
A + (B + C) = (A + B) + C
Commutative law
A + B = B + A
Scalar Multiplication
k(A) = kA
where k is a scalar quantity

Negative of a vector and vector subtraction
                    The negative of a vector is simply a vector with the same magnitude, but pointed in the opposite direction (see figure below).

To perform vector subtraction, follow the same rules as vector addition, the only difference being the addition of a vector oriented in the opposite direction as in the figure below.
The resultant is also drawn from the tail of the first vector to the tip of the last vector.
Vector Components and Unit Vectors
Coordinate Systems
A Cartesian coordinate system is shown in the figure below. The coordinate system is broken into quadrants represented by: I, II, III, IV in the diagram.  I = (+,+), II = (-,+), III = (-,-), IV = (+,-)
Algebraic Addition/Subtraction of Vectors
Magnitude and Direction of Vectors
Consider a vector R = (Rx, Ry), the magnitude and angle are found the following way:
|R| = [(Rx)2 + ( Ry)2]1/2
q = tan-1(Ry/Rx)
Vector Components
If q   is the angle that R makes with the x-axis, then
Rx = |R|cosq
Ry = |R|sinq
Addition/Subtraction of vectors:
Suppose we have 2 vectors  A and B such that
                    C = A + B
                   Cx = (Ax +Bx)
                    Cy = (Ay +By)

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