SCALARS AND VECTORS REPRESENTING VECTORS

HOW DO WE DEFINE SCALAR AND VECTOR QUANTITIES? Scalar Quantity – a quantity represented only by its magnitude.  Jogging 10 km Mass, time, distance, speed, work, energy Vector Quantity – a quantity represented not just by its magnitude, but also its direction  Jogging 5 km, North then 5 km back. Displacement, velocity, acceleration, force, weight, momentum

You can reach him by caravan, or cross the desert like an Arab man, I don’t care how you get there, just get there if you can! But there are hills and mountains. Take the golden can, put it in a tan van, 10 km east, then take it to Dan, 5 km north, then to Fran, 2 km northeast, and Stan, 1 km west. Got it man? Yes, Mr. D!

TAIL-TIP METHOD Tail Head

THE CARTESIAN GRAPH Quadrant I (+,+) Quadrant II (-,+) Quadrant III (-,-) Quadrant IV (+,-) NORTH EASTWEST SOUTH +y-axis +x-axis-x-axis -y-axis Graphing is a way to display numerical relationships as pictures. With pictures, it is easier to see patterns in numbers. A philosopher and mathematician from the early 17th century named Renes Descartes was the first to use the coordinate system we use today. The system is named the Cartesian Coordinate System, after him. This is the Cartesian Plane. The horizontal axis is called the x-axis. The vertical axis is the y-axis.

MORE EXAMPLES 1.) x - component = -4, y - component 2.) magnitude = 6, direction = SOUTH 3.) magnitude = 7, direction = SOUTH EAST 4.) x – component = 0, y – component = -7

WHY FOCUS ON VECTOR ADDITION? WHAT ABOUT SCALARS? Scalar mathematics is ordinary mathematics. This means that if you want to add up 4m and 6m, you simply add them: 4m + 6m = 10m. No more, no less. In dealing with vectors however, their directional property should be taken into account. That is why the mathematical treatment to which we subject vectors is different from the arithmetic we have grown accustomed to is not always equal to 4 in vector mathematics. “For the philosophers”: “A vector with a magnitude of 2 and another vector with a magnitude of 2, when added, can be equal to zero (2+2 = 0!)

NEGATIVE VECTOR A vector that has the same magnitude as the given vector, but opposite in direction. A - A We can now add the negative vector A to the other Vector B. So B – A + B + (-A)

RESULTANT VECTOR The vector resulting from two or more vectors. When adding vectors, any non zero result will also be a vector. What comes out is what we call the resultant vector, or simply resultant. Graphical addition of vectors is performed on a neat, accurate, scale diagram. This concept implies that all vectors present in an given system can be substituted by a single vector, the resultant vector.

DISTANCE AND DISPLACEMENT

Distance and displacement are two quantities that may seem to mean the same thing yet have distinctly different definitions and meanings. Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion. Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position. To test your understanding of this distinction, consider the motion depicted in the diagram below. A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North.

EXPLANATION Even though the physics teacher has walked a total distance of 12 meters, her displacement is 0 meters. During the course of her motion, she has "covered 12 meters of ground" (distance = 12 m). Yet when she is finished walking, she is not "out of place" - i.e., there is no displacement for her motion (displacement = 0 m). Displacement, being a vector quantity, must give attention to direction. The 4 meters east cancels the 4 meters west; and the 2 meters south cancels the 2 meters north. Vector quantities such as displacement are direction aware. Scalar quantities such as distance are ignorant of direction. In determining the overall distance traveled by the physics teachers, the various directions of motion can be ignored.

EXAMPLE 1 Now consider another example. The diagram below shows the position of a cross-country skier at various times. At each of the indicated times, the skier turns around and reverses the direction of travel. In other words, the skier moves from A to B to C to D. Use the diagram to determine the resulting displacement and the distance traveled by the skier during these three minutes.

ANSWER Now consider another example. The diagram below shows the position of a cross- country skier at various times. At each of the indicated times, the skier turns around and reverses the direction of travel. In other words, the skier moves from A to B to C to D. Use the diagram to determine the resulting displacement and the distance traveled by the skier during these three minutes. The skier covers a distance of (180 m m m) = 420 m and has a displacement of 140 m, rightward.

EXAMPLE 2 consider a football coach pacing back and forth along the sidelines. The diagram below shows several of coach's positions at various times. At each marked position, the coach makes a "U-turn" and moves in the opposite direction. In other words, the coach moves from position A to B to C to D. What is the coach's resulting displacement and distance of travel?