1. Operations on Vectors

2. 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

3. Tip to Tail Method u v

4. Parallelogram Method u v

5. Vector Subtraction Property of Subtraction  u - v = u + (-v)

6. Calculating the norm and direction of resultant vectors Pythagorean Theorem  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. θ 7m 5m R c 2 = a 2 + b 2 c 2 = 5 2 + 7 2 c 2 = 74 c = 8.6 m Tanθ = 7/5 =1.4 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. 35 m, E 20 m, N 12 m, W 6 m, S - = 23 m, E -= 14 m, N 23 m, E 14 m, N The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST R 

9. Example Tommy travels 5 km North and then decides to travel 3.9km [W 5°N]. Determine the vector that represents the distance and orientation from his starting point.

10. Example Determine the resultant vector of u - v u v 50° 60° 3 cm 50° 60° 3 cm 50° 3 cm R

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

12. Example Simplify each expression: a) CD + DE + EF b) AB – FB c) -CD + CE - FE

13. Multiplication of a Vector by a Scalar The product of a non-zero vector and a scalar is 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

15. 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)

16. Example Consider the following vectors: u = (8,4)v = (2,1)w = (6,-2) Calculate a) u + v b) w – vc) 3u – v + 2w d) ll 5w – u ll

17. Example 1. Draw vector v if (- 4v) is represented below. 2. Reduce: -2u + v – 6v + 3u