1 / 17

Operations on Vectors

Operations on Vectors. Vector Addition. There are two methods to add vectors u and v Tip to tail (triangle method ) Parallelogram Properties of Addition u + v = v + u (u + v) + w = u + (v + w) u + 0 = u u + (-u) = 0. u. v. Tip to Tail Method. u. v.

mura
Download Presentation

Operations on Vectors

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. Operations on Vectors

  2. Vector Addition • There are twomethods to addvectorsu and v • Tip to tail (triangle method) • Parallelogram • Properties of Addition • u + v = v + u • (u + v) + w = u + (v + w) • u + 0 = u • u + (-u) = 0 u v

  3. Tip to Tail Method u v

  4. Parallelogram Method u v

  5. VectorSubtraction • Property of Subtraction • u - v = u + (-v)

  6. Calculating the norm and direction of resultantvectors • PythagoreanTheorem • Right angle triangles only! • Sine Law • Cosine Law

  7. Examples Tony walks 5m West and 7m North. Determine the length and angle of the resultant motion. c2 = a2 + b2 c2 = 52 + 72 R c2 = 74 7m c = 8.6 m θ Tanθ= 7/5 =1.4 5m θ= 54.5° 8.6m W 54.5° N

  8. Example A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 23 m, E - = 12 m, W - = 14 m, N 6 m, S 20 m, N 14 m, N 35 m, E R q 23 m, E The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST

  9. Example Tommy travels 5 km North and thendecides to travel 3.9km [W 5°N]. Determine the vectorthatrepresents the distance and orientation fromhisstarting point.

  10. Example • Determine the resultantvector of u- v 3 cm u 3 cm 60° 60° R 50° 50° v 3 cm 3 cm 50°

  11. Chasles Relation • If A, B and C are three points in a cartesian plane, then: AB + BC = AC B A C

  12. Example • Simplifyeach expression: • CD + DE + EF • AB – FB • -CD + CE - FE

  13. Multiplication of a Vector by a Scalar The product of a non-zerovector and a scalaris a vector au if a > 0, u and au same direction if a < 0, u and au opposite directions • Properties of Multiplication • a(bu) = ab(u) • u x 1 = u • a(u + v) = au + av

  14. Algebraic Vectors Operations between Algebraic Vectors Given vectors u = (a,b) and v = (c,d) u + v = (a + c, b + d) u - v = (a - c, b - d) ku = (ka,kb) where k is any real number (scalar)

  15. Example Consider the followingvectors: u = (8,4) v = (2,1) w = (6,-2) Calculate a) u + v b) w – v c) 3u – v + 2w d) ll 5w – ull

  16. Example 1. Drawvectorv if (- 4v)isrepresentedbelow. 2. Reduce: -2u + v – 6v + 3u

More Related