Vectors • Recall that “vectors” are arrows that represent a vector quantity (magnitude and direction). • The length of the arrow represents the magnitude of a measurement • The direction of the arrow within a coordinate system represents the direction of the vector. • Most vector directions are references to a particular direction, i.e., 0o or say the “west” or negative x-axis of a graph.
Vectors move! • Vector quantities of the same units can be added to each other. • Vectors can be added in 2 ways: • Geometrically • Algebraically • To add 2 vectors geometrically simply place the tail of one vector to the head of the other. The solution, or Resultant vector, is the vector from the point of origin of the 1st vector to the end point of the 2nd. • When adding vectors geometrically, one often places the first vector at the origin.
Example: Fb Fa Fb R Fa Fb R R Fa R = magnitude R @ o North of East
Adding Vectors Algebraically • Adding vectors algebraically involves simply adding magnitudes that are along the same axes. The Unit Vector • A unit vector is a vector of 1 “unit” in length that defines a particular direction. • For example: In a Cartesian system, there are 3 principal axes: x, y, & z. The unit vector of each is simply a vector of length 1, in each direction.
Often a vector may be written as the sum of its parts, each multiplied by a unit vector giving the direction associated with that part. The vector R shown at the right could be written as R = 3x + 4y Notice that the vector R can be Represented as the geometric sum of 3 times the x-unit vector plus 4 times The y-unit vector. y R 4 x 3
Notation • i, j, k notation • Arrows • Boldface
Algebraic Sums • The resultant vector obtained from the graphical addition of two vectors can be found form adding the vectors algebraically. 1st: Break each vector into independent components. I.e., Put the vector into i,j,k notation if it’s not already. 2nd: Add each component of both vectors independently of the other components.
Example • A = 3i + 4j +5k • B = 2i +7k R = A + B R = 3i + 4j +5k + 2i +7k OR 3i + 4j +5k + 2i + 0j +7k R = 5i + 4j + 12k
Resolution of Vectors into Components • Vectors describing real conditions are rarely written in the easy-to-use Cartesian notation. More often a vector is expressed as an angle. • For example, the velocity of a projectile might be given as 30 m/s at 25o above the plain.
In order to work with this velocity vector in a meaningful context we must often “resolve” it, or break it down into its component parts. • Generally we will employ the use of trigonometry to accomplish this task.
Angles • To find the angle formed when two independent (i.e., x- and y- dimension) vectors are added simply use the ratio of their magnitudes and the tangent function.
Independent motion • Since we are using vectors as variables to analyze motion it is critical to note that vectors that are at right angles to each other are independent. • This means that motion in any one direction does not affect motion at right angles to that referenced direction. • For example: Up/Down (Vertical) motion is independent of Side/Side (Horizontal) motion during projectile flight.
Products of Vectors • The Scalar Product • The “dot” product The scalar product of two vectors is the product of the magnitude of one vector and the component of a second vector along the direction of the first. The scalar or dot product takes the form:
The Vector Product(Cross Product) • The cross product of two vectors is given by where is the smallest angle between the two vectors. • The direction of the resultant vector is perpendicular to the plane defined by vectors A & B and given by the right hand rule.