Geom9point7
- 1. Chapter 2 - Vectors
- 2. Objectives Understand vectors and their components on the coordinate system Understand vector addition by the parallelogram method and by the component method Understand vectors in a state of equilibrium
- 3. Vectors on the Coordinate System Horizontal, vertical, and slanted vectors can be drawn on the coordinate system. All 3 types of vectors have both length and direction. All 3 types can be represented by signed numbers.
- 4. Slanted Vectors The direction of slanted vectors is stated in terms of The angle formed by the vector and the horizontal axis. The quadrant in which that angle is formed. The length of this vector is ___? The direction of this vector is a ___ angle in the ___ quadrant. 3 units Θ = 40˚
- 5. Slanted Vectors The angle which specifies the direction of a slanted vector is called its reference angle. All slanted vectors have positive lengths. Vectors are named using 2 letters: AB The first letter of the name is always where the vector begins. 3 units Θ = 40˚ A B
- 6. Slanted Vectors Any slanted vector has a horizontal and a vertical component. We can calculate these because we can make this a right triangle and use trig. 3 units Θ = 40˚ A B
- 7. Slanted Vectors How do we calculate the horizontal component (AC)? Cos θ = adj/hyp = x/3 .7660 = x/3 X = 3 * .7660 = 2.298 sin θ = opp/hyp = x/3 .6428 = x/3 X = 3 * .7660 = 1.9284 Use Pyth to check 2.298 2 + 1.9284 2 ?=? 3 2 3 units Θ = 40˚ A B C
- 8. Try some of these Do p. 81 #s 54-62
- 9. Flipping the problem Tan = opp/adj Tan θ = 4/5 = .8000 Therefore θ contains 39˚ Pyth can help us find the length of AB: AB 2 = AC 2 +BC 2 AB = 5 2 + 4 2 AB = 25 + 16 = 41 AB = 6.4 How would you do this using sin and cos? Θ = ?˚ A B = (5, 4) C
- 10. Adding Vectors What does it mean to add two vectors? Vector and Field (vector addition) Why do we care? Canoe A= 5,2 0 C = -4,3 θ
- 12. Adding Vectors In Physics, the Law of Conservation and Momentum uses this. Now how do we do that without the website? Create a parallelogram and find the diagonal. A= 5,2 0 C = -4,3 θ
- 13. Adding Vectors Draw AQ which is both parallel to OC and equal in length to OC. Draw CQ which is both parallel to OA and equal in length to OA On a graph, we can see that the points of Q are 2,6 A= 5,2 0 C = -4,3 θ Q
- 14. Adding Vectors We can draw one line, then a vector from the origin to point Q: 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. Would this be precise if we weren’t using whole numbers? A= 5,2 0 C = -4,3 θ Q
- 15. Adding Vectors Another way to add vectors is by the component method. This provides accurate answers without the necessity of constructing parallelograms. Find the horizontal and vertical components, and add them Horizontal: -3 + 5 =2 Vertical: 4 + 2 = 6 A= 5,2 0 C = -3, 4 θ Q
- 16. Vector addition Positives and negatives are extremely important – be careful with them. A= 5,2 0 C = -4,3 θ Q
- 17. Vector addition To find the length and direction of the resultant vector, we use trig. Use Pyth to find the length of OC Use tan to find the reference angle of OC C= 25.6, 12.7 0 F = -7.9, 7.2 α Q
- 18. Applications How is vector addition used in physics?
- 20. Homework Do page 576 10-30 evens
- 21. Adding Vectors on the Coordinate Axes On page 107, #103, the vertical components are both 0 And the reference angle is 0 ˚ So adding these is pretty easy! On #104, the horizontal components are both 0 And the reference angle is 90 ˚ Once again, addition is easy
- 22. Finding a Vector-Addend The sum of a vector addition is called the resultant. The two vectors which are added are called vector addends. Use Pythagorean and tan
- 23. Adding more than two vectors The parallelogram method doesn’t work – use the component method See example at the bottom of page 116.
- 24. Vector Opposites, the Zero-Vector, and the State of Equilibrium Two numbers are a pair of opposites if their sum is 0. Two vectors are a pair of vector opposites if both The sum of their horizontal components is 0 The sum of their vertical components is 0 See example on p. 119 #122 When the resultant of two vectors is the zero vector, we say that the two vectors are in a state of equilibrium. Can you think of an example of this in real life?
- 25. The Zero-Vector What is the horizontal component of the zero vector? 0 What is the vertical component of the zero vector? 0 What is the length of the zero vector? 0
- 26. Equilibrants and the State of Equilibrium If a system of vectors is not in a state of equilibrium, we can always add one more vector which produces a state of equilibrium. This added vector is called an equilibrant. If 3 vectors are in a state of equilibrium, each vector is the vector-opposite of the resultant of the other 2 vectors. See examples on p. 123
- 27. Homework Do Self-Tests 9, 10, 11, 12 & 13