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Directions – Pointed Things

This lesson covers the basics of vectors, including kinematics and direction. Learn how to add and subtract vectors graphically and multiply/divide vectors by scalars. Get ready for quizzes with trigonometry and bring a calculator!

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Directions – Pointed Things

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  1. Directions – Pointed Things January 23, 2005 Vectors

  2. Today … • We complete the topic of kinematics in one dimension with one last simple (??) problem. • We start the topic of vectors. • You already read all about them … right??? • There is a WebAssign problem set assigned. • The exam date & content may be changed. • Decision on Wednesday • Study for it anyway. “may” and “will” have different meanings! Vectors

  3. One more Kinematics Problem A freely falling object requires 1.50 s to travel the last 30.0 m before it hits the ground. From what height above the ground did it fall? h=?? 30 meters 1.5 seconds Vectors

  4. The Consequences of Vectors • You need to recall your trigonometry if you have forgotten it. • We will be using sine, cosine, tangent, etc. thingys. • You therefore will need calculators for all quizzes from this point on. • A cheap math calculator is sufficient, • Mine cost $11.00 and works just fine for most of my calculator needs. Vectors

  5. Hey … I’m lost. How do I get to lake Eola NO PROBLEM! Just drive 1000 meters and you will be there. B HEY! Where the *&@$ am I?? A Vectors

  6. The problem: The moron drove in the wrong direction! Vectors

  7. IS this the way (with direecton now!) to Lake Eola? B THIS IS A VECTOR A Vectors

  8. YUM!! Vectors

  9. The following slides were supplied by the publisher ……………… thanks guys! Vectors

  10. Vectors and Scalars • A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. • A vector quantity is completely described by a number and appropriate units plus a direction. Vectors

  11. Vector Notation • When handwritten, use an arrow: • When printed, will be in bold print: A • When dealing with just the magnitude of a vector in print, an italic letter will be used: A or |A| • The magnitude of the vector has physical units • The magnitude of a vector is always a positive number Vectors

  12. Vector Example • A particle travels from A to B along the path shown by the dotted red line • This is the distance traveled and is a scalar • The displacement is the solid line from A to B • The displacement is independent of the path taken between the two points • Displacement is a vector Vectors

  13. Equality of Two Vectors • Two vectors are equal if they have the same magnitude and the same direction • A = B if A = B and they point along parallel lines • All of the vectors shown are equal Vectors

  14. Coordinate Systems • Used to describe the position of a point in space • Coordinate system consists of • a fixed reference point called the origin • specific axes with scales and labels • instructions on how to label a point relative to the origin and the axes Vectors

  15. Cartesian Coordinate System • Also called rectangular coordinate system • x- and y- axes intersect at the origin • Points are labeled (x,y) Vectors

  16. Polar Coordinate System • Origin and reference line are noted • Point is distance r from the origin in the direction of angle , ccw from reference line • Points are labeled (r,) Vectors

  17. Polar to Cartesian Coordinates • Based on forming a right triangle from r and q • x = r cos q • y = r sin q Vectors

  18. Cartesian to Polar Coordinates • r is the hypotenuse and q an angle • q must be ccw from positive x axis for these equations to be valid Vectors

  19. Example • The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point. • Solution: From Equation 3.4, and from Equation 3.3, Vectors

  20. If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are (r, 30), determine y and r. Vectors

  21. Adding Vectors Graphically Vectors

  22. Adding Vectors Graphically Vectors

  23. Adding Vectors, Rules • When two vectors are added, the sum is independent of the order of the addition. • This is the commutative law of addition • A + B = B + A Vectors

  24. Adding Vectors • When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped • This is called the Associative Property of Addition • (A + B) + C = A + (B + C) Vectors

  25. Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, find (a) the vector sum A + B and (b) the vector difference AB. Vectors

  26. Subtracting Vectors • Special case of vector addition • If A – B, then use A+(-B) • Continue with standard vector addition procedure Vectors

  27. Multiplying or Dividing a Vector by a Scalar • The result of the multiplication or division is a vector • The magnitude of the vector is multiplied or divided by the scalar • If the scalar is positive, the direction of the result is the same as of the original vector • If the scalar is negative, the direction of the result is opposite that of the original vector Vectors

  28. Each of the displacement vectors A and B shown in Fig. P3.15 has a magnitude of 3.00 m. Find graphically (a) A + B, (b) A  B, (c) B  A, (d) A  2B. Report all angles counterclockwise from the positive x axis. Vectors

  29. Components of a Vector • A component is a part • It is useful to use rectangular components • These are the projections of the vector along the x- and y-axes Vectors

  30. Unit Vectors • A unit vector is a dimensionless vector with a magnitude of exactly 1. • Unit vectors are used to specify a direction and have no other physical significance Vectors

  31. Unit Vectors, cont. • The symbols represent unit vectors • They form a set of mutually perpendicular vectors Vectors

  32. Unit Vectors in Vector Notation • Ax is the same as AxandAy is the same as Ayetc. • The complete vector can be expressed as Vectors

  33. Adding Vectors with Unit Vectors Vectors

  34. Adding Vectors Using Unit Vectors – Three Directions • Using R = A + B • Rx = Ax + Bx , Ry = Ay+ By and Rz = Az + Bz etc. Vectors

  35. Consider the two vectors and . Calculate (a) A + B, (b) AB, (c) |A + B|, (d) |AB|, and (e) the directions of A + B and AB . Vectors

  36. Three displacement vectors of a croquet ball are shown in the Figure, where |A| = 20.0 units, |B| = 40.0 units, and |C| = 30.0 units. Find (a) the resultant in unit-vector notation and (b) the magnitude and direction of the resultant displacement. Vectors

  37. The helicopter view in Fig. P3.35 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown, and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (abbreviated N). Vectors

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