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
Parallelogram Method u v
Vector Subtraction Property of Subtraction u - v = u + (-v)
Calculating the norm and direction of resultant vectors Pythagorean Theorem Right angle triangles only! Sine Law Cosine Law
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 = c 2 = 74 c = 8.6 m Tanθ = 7/5 =1.4 θ = 54.5° 8.6m W 54.5° N
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: m, 31.3 degrees NORTH of EAST R
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.
Example Determine the resultant vector of u - v u v 50° 60° 3 cm 50° 60° 3 cm 50° 3 cm R
Chasles Relation If A, B and C are three points in a cartesian plane, then: AB + BC = AC A B C
Example Simplify each expression: a) CD + DE + EF b) AB – FB c) -CD + CE - FE
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
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)
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
Example 1. Draw vector v if (- 4v) is represented below. 2. Reduce: -2u + v – 6v + 3u