1 / 32

Mastering Vector Formulas and Graphs in Polar Equations

Explore vector formulas, unit vectors, dot products, and graphs in polar equations. Learn how to prove orthogonality, parallelism, and analyze polar graphs for symmetry, petals, and curves.

tiger
Télécharger la présentation

Mastering Vector Formulas and Graphs in Polar Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6 Review Due 5/21 # 2 – 22 even # 53 – 59 odd # 62 – 70 even # 74, 81, 86 (p. 537)

  2. Vector Formulas Unit Vectors: Horizontal/Vertical components: Angle between Vectors: Projections:

  3. 6.1 Vectors in a Plane Day # 1

  4. RS starts at R and goes to S v = direction magnitude (size) force acceleration velocity Starts at (0, 0) and goes to (x, y)

  5. v B A AB v = equivalent

  6. Q P

  7. Vector addition Vector multiplication (multiplying a vector by a scalar or real number) sum terminal initial point point parallelogram law

  8. unit vector unit vector direction

  9. direction angle

  10. 25o

  11. 6.2 Dot Product of Vectors Day # 1

  12. dot product work done vectors scalar (real number)

  13. Theorem: Angles Between Vectors If θ is the angle between the nonzero vectors u and v, then

  14. Prove that the vectors are orthagonal: Proving Vectors are Orthagonal

  15. Prove that the vectors are parallel: Proving Vectors are Parallel The vectors u and v are parallel if and only if: u = kv for some constant k

  16. Show that the vectors are neither: Proving Vectors are Neither If 2 vectors u and v are not orthagonal or parallel: then they are NEITHER

  17. vector projection

  18. Unit Circle

  19. 6.4 Polar Equations Day # 1

  20. P r θ O polar axis pole polar coordinate system polar axis ( r, θ ) polar coordinates directed distance polar axis directed angle line OP

  21. P(r, θ) r y θ x Cartesian (rectangular) Polar pole origin polar axis positive x – axis x = r cos θ y = r sin θ

  22. P(x, y) r y θ x so so

  23. Helpful Hints Polar to Rectangular multiply cos or sin by r so you can convert to x or y r2 = x2 + y2 re-write sec and csc as complete the square as necessary Rectangular to Polar replace x and y with rcos and rsin when given a “squared binomial”, multiply it out x2 + y2 = r2 (x – a)2 + (y – b)2 = c2 Where the center of the circle is (a, b) and the radius is c

  24. 6.5 Graphs of Polar Equations Day # 1

  25. General Form: r = a cos n θ r = a sin n θ Petals: n: odd  n petals n: even  2n petals n: even n: odd cos  one petal on pos. x-axis cos  petals on each side of each axis sin  one petal on half of y-axis sin  no petals on axes

  26. General Form: r = a + b sin θ r = a + b cos θ Symmetry: sin: about y – axis cos: about x – axis when , there is an “inner loop” (#5) when , it touches the origin; “cardioid” (#6) when , it’s called a “dimpled limacon” (#7) when , it is a “convex limacon” (#8)

  27. ANALYZING POLAR GRAPHS • We analyze polar graphs much the same way we do graphs of rectangular equations. • The domain is the set of possible inputs for . The range is the set of outputs for r. The domain and range can be read from the “trace” or “table” features on your calculator. • We are also interested in the maximum value of. This is the maximum distance from the pole. This can be found using trace, or by knowing the range of the function. • Symmetry can be about the x-axis, y-axis, or origin, just as it was in rectangular equations. • Continuity, boundedness, and asymptotes are analyzed the same way they were for rectangular equations.

  28. What happens in either type of equation when the constants are negative? Draw sketches to show the results. • Rose Curve when “a” is negative (“n” can’t be negative, by definition) • if n is even, picture doesn’t change…just the order that the points are plotted changes • if n is odd, the graph is reflected over the x – axis

  29. What happens in either type of equation when the constants are negative? Draw sketches to show the results. • Rose Curve when “a” is negative (“n” can’t be negative, by definition) • if n is even, picture doesn’t change…just the order that the points are plotted changes • if n is odd, the graph is reflected over the y – axis

  30. What happens in either type of equation when the constants are negative? Draw sketches to show the results. • Limacon Curve when “b” is negative (minus in front of the b) (“a” can’t be negative, by definition) • when r = a + bsinθ, the majority of the curve is around the positive y – axis. • when r = a – bsinθ, the curve flips over the x – axis.

  31. What happens in either type of equation when the constants are negative? Draw sketches to show the results. • Limacon Curve when “b” is negative (minus in front of the b) (“a” can’t be negative, by definition) • when r = a + bcos θ, the majority of the curve is around the positive x – axis. • when r = a – bcos θ, the curve flips over the y – axis.

More Related