Geom9point7 97

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Geom9point7 97

  1. 1. Chapter 2 - Vectors
  2. 2. Objectives <ul><li>Understand vectors and their components on the coordinate system </li></ul><ul><li>Find the magnitude of a vector </li></ul><ul><li>Understand vector addition by the parallelogram method and by the component method </li></ul><ul><li>Understand vectors in a state of equilibrium </li></ul>
  3. 3. Vectors on the Coordinate System <ul><li>Horizontal, vertical, and slanted vectors can be drawn on the coordinate system. </li></ul><ul><li>All 3 types of vectors have both length and direction. </li></ul>
  4. 4. Slanted Vectors <ul><li>The direction of slanted vectors is stated in terms of </li></ul><ul><ul><li>The angle formed by the vector and the horizontal axis. </li></ul></ul><ul><ul><li>The quadrant in which that angle is formed. </li></ul></ul><ul><li>The length of this vector is ___? </li></ul><ul><li>The direction of this vector is a ___ angle in the ___ quadrant. </li></ul>3 units Θ = 40˚
  5. 5. Slanted Vectors <ul><li>The angle which specifies the direction of a slanted vector is called its reference angle. </li></ul><ul><li>All slanted vectors have positive lengths. </li></ul><ul><li>Vectors are named using 2 letters: </li></ul><ul><ul><li>AB </li></ul></ul><ul><li>The first letter of the name is always where the vector begins. </li></ul>3 units Θ = 40˚ A B
  6. 6. Slanted Vectors <ul><li>Any slanted vector has a horizontal and a vertical component. </li></ul><ul><li>We can calculate these because we can make this a right triangle and use trig. </li></ul>3 units Θ = 40˚ A B
  7. 7. Magnitude <ul><li>How long is this vector? </li></ul><ul><li>Use the distance formula! </li></ul><ul><li>If A(x 1 , y 1 ) and B(x 2 , y 2 ) are points in a coordinate plane, then the distance between A and B is </li></ul><ul><li>AB = √(x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 </li></ul><ul><li>AB = (4-0) 2 + (5-0) 2 </li></ul><ul><li>AB = √ 16 + 25 </li></ul><ul><li>AB = √41 </li></ul>4,5 0,0 A B
  8. 8. Magnitude <ul><li>Try another one: </li></ul><ul><li>AB = √(x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 </li></ul><ul><li>AB = √(5-0) 2 + (4-2) 2 </li></ul><ul><li>AB 2 = √ 25 + 4 </li></ul><ul><li>AB = √29 = 5.4 </li></ul>5,4 0,2 A B
  9. 9. Component Form <ul><li>The component form of a vector is written as <x, y> where x is ( x 2 - x 1 ) and y is y 2 – y 1 </li></ul><ul><li>What is the component form of this vector? </li></ul><ul><li><(5-0), (4-2)> </li></ul><ul><li><5, 2> </li></ul>5,4 0,2 A B
  10. 10. Another example <ul><li>What is the component form of this vector? </li></ul><ul><li><4, 5> </li></ul>4,5 0,0 A B
  11. 11. Direction <ul><li>The direction of a vector is determined by the angle it makes with the horizontal line. </li></ul><ul><li>What direction is vector AB heading? </li></ul><ul><li>If AB represents the velocity of a moving ship, and the scale on the axis is miles per hour, how fast is the ship moving? </li></ul>3,4 0,0 A B
  12. 12. Equal and Parallel Vectors <ul><li>Two vectors are equal if they have the same magnitude and direction. </li></ul><ul><li>Two vectors are parallel if they have the same or opposite directions. </li></ul>3,4 0,0 A B
  13. 13. Slanted Vectors <ul><li>How do we calculate the horizontal component (AC)? </li></ul><ul><li>Cos θ = adj/hyp = x/3 </li></ul><ul><li>.7660 = x/3 </li></ul><ul><li>X = 3 * .7660 = 2.298 </li></ul><ul><li>sin θ = opp/hyp = x/3 </li></ul><ul><li>.6428 = x/3 </li></ul><ul><li>X = 3 * .7660 = 1.9284 </li></ul><ul><li>Use Pyth to check </li></ul><ul><li>2.298 2 + 1.9284 2 ?=? 3 2 </li></ul>3 units Θ = 40˚ A B C
  14. 14. Flipping the problem <ul><li>Tan = opp/adj </li></ul><ul><li>Tan θ = 4/5 = .8000 </li></ul><ul><li>Therefore θ contains 39˚ </li></ul><ul><li>Pyth can help us find the length of AB: </li></ul><ul><li>AB 2 = AC 2 +BC 2 </li></ul><ul><li>AB = 5 2 + 4 2 </li></ul><ul><li>AB = 25 + 16 = 41 </li></ul><ul><li>AB = 6.4 </li></ul><ul><li>How would you do this using sin and cos? </li></ul>Θ = ?˚ A B = (5, 4) C
  15. 15. Adding Vectors <ul><li>What does it mean to add two vectors? </li></ul><ul><li>Vector and Field (vector addition) </li></ul><ul><li>Why do we care? Using Vectors Video </li></ul>A= 5,2 0 C = -4,3 θ
  16. 16. Adding Vectors <ul><li>In Physics, the Law of Conservation and Momentum uses this. </li></ul><ul><li>Now how do we do that without the website? </li></ul><ul><li>Create a parallelogram and find the diagonal. </li></ul>A= 5,2 0 C = -4,3 θ
  17. 17. Adding Vectors <ul><li>Draw AQ which is both parallel to OC and equal in length to OC. </li></ul><ul><li>Draw CQ which is both parallel to OA and equal in length to OA </li></ul><ul><li>On a graph, we can see that the points of Q are 2,6 </li></ul>A= 5,2 0 C = -4,3 θ Q
  18. 18. Adding Vectors <ul><li>We can draw one line, then a vector from the origin to point Q: </li></ul><ul><li>This lets us find the point on graph paper without a calculator. Even using a calculator, this is a nice way to prove we’re doing things correctly. </li></ul><ul><li>Would this be precise if we weren’t using whole numbers? </li></ul>A= 5,2 0 C = -4,3 θ Q
  19. 19. Adding Vectors <ul><li>Another way to add vectors is by the component method. </li></ul><ul><ul><li>This provides accurate answers without the necessity of constructing parallelograms. </li></ul></ul><ul><li>Find the horizontal and vertical components, and add them </li></ul><ul><li>Horizontal: -3 + 5 =2 </li></ul><ul><li>Vertical: 4 + 2 = 6 </li></ul>A= 5,2 0 C = -3, 4 θ Q
  20. 20. Vector addition <ul><li>Positives and negatives are extremely important – be careful with them. </li></ul>A= 5,2 0 C = -4,3 θ Q
  21. 21. Vector addition <ul><li>To find the length and direction of the resultant vector, we use trig. </li></ul><ul><li>Use Pyth to find the length of OC </li></ul><ul><li>Use tan to find the reference angle of OC </li></ul>C= 25.6, 12.7 0 F = -7.9, 7.2 α Q
  22. 22. Applications <ul><li>How is vector addition used in physics? </li></ul><ul><li>Law of Conservation Video </li></ul>

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