Scalar Quantities Scalar quantities are measurements that have only magnitude (size) but no direction. Examples of scalar quantities are; Distance Speed Temperature

Vector Quantities Vector quantities are measurements which have both magnitude (size) and direction. Examples of vector quantities are; Displacement Velocity Acceleration Force

Displacement (a vector) vs. Distance (a scalar) If you leave your driveway and run a bunch of errands and then return and park in the same spot, what is you distance and what is your displacement? Your distance was recorded on your odometer. Let’s say it is 10 miles. Your displacement is the distance from where you started to where you finished. In this case it is zero miles.

Speed vs. Velocity Speed and velocity are used interchangeably but they are not the same thing. Speed is only how fast you are going with no direction. Velocity is how fast you are going and in what direction.

Coordinate Systems You need to remember the X-Y coordinate system into which a graph is divided into four quarters by a vertical and horizontal lines. Each quadrant contains 90 o and if you add all four quadrants you get 360 o (a full circle).

Up = +Down = -Right = + Left = - y x Quadrant IQuadrant II Quadrant IIIQuadrant IV 0 o East 90 o North West 180 o 270 o South 360 o Rectangular Coordinates

Coordinate Systems Note: That there are multiple ways of indicating each direction. East or Right which is + West or Left which is – North or Upward which is + South or Downward which is – Also note that the measure of degrees starts at the East line and moves counterclockwise around the graph.

315 O 0 O East 90 O North West 180 O 270 O South 360 O +x +y - x - y O or 45 O SOUTH OF EAST VECTOR NOTATIONS

Polar Form Example Polar Form: You find the magnitude of the new vector line by measuring the length on the graph. For our example this gives; c = 14.1

Polar Form Example The angle can be determined using the graph. Angle = 315 o or 45 o South of East

Adding Vectors Graphically Suppose you have two people pushing on a box. If they are pushing on the same side with an equal force, you can add the two forces together to get the total force on the box.

Adding Vectors Graphically Suppose now these two people are pushing on an object that is irregular in shape. One might be pushing a little to the left and the other a little to the right. Determining the total force has become more difficult since the two people are pushing against each other at the same time that they are pushing against the object.

Adding Vectors Graphically One method of figuring out the force (or a number of other factors) is to add the two vectors representing these forces graphically.

Adding Vectors Graphically You assign each inch or centimeter or some other measurement the value of 100 mph. Then you draw a line 5 inches in one direction and 3 inches in the other.

Adding Vectors Graphically The solution (called the resultant) is determined by drawing a new line from the tail of the first vector to the head of the second vector.

Adding Vectors Graphically As the bottom illustration shows you can repeat this process for any number of vectors. Each must be drawn to scale and oriented in the correct direction. Note that you can graphically add two vectors that are not perpendicular (at a right angle of 90 o ) graphically easier than doing it mathematically.

Adding Vectors Graphically Once the resultant is drawn, you measure the length of the resultant and translate that into you answer using the scale that you set up to draw the vectors you are adding.

Adding Vectors Graphically However, with a vector you need a direction as well as a magnitude. You measure the direction of the resultant line using a protractor. You report the angle, you reference the angle to one of the for axis (North, South, East or West).

Adding Vectors Graphically Example: You have two vectors. One is 12 N of force acting at 20 o North of East. The second is 25 N acting at 30 o East of North. What is the resultant force?

Adding Vectors Graphically