Vectors: Introduction
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1 Vectors: Introduction Calculus II Josh Engwer TTU 23 April 2014 Josh Engwer (TTU) Vectors: Introduction 23 April / 30
2 Vectors & Scalars (Definition) Definition A vector v is a quantity that bears both magnitude and direction. Examples of vectors: displacement, velocity, force, angular momentum,... Definition A scalar t is a quantity that bears only magnitude. Examples of scalars: time, temperature, distance, speed, area, volume,... Josh Engwer (TTU) Vectors: Introduction 23 April / 30
3 2D Vector (Definition) Vector v with horizontal component v 1 and vertical component v 2 : Josh Engwer (TTU) Vectors: Introduction 23 April / 30
4 2D Vector (Definition) Vector PQ with initial point P and terminal point Q: Josh Engwer (TTU) Vectors: Introduction 23 April / 30
5 2D Vector (Definition) Vector QP with initial point Q and terminal point P: Josh Engwer (TTU) Vectors: Introduction 23 April / 30
6 2D Vector (Representation) 2D Space: R 2 (say R-Two ) Component Form: v = v 1, v 2 Basis Form: v = v 1 î + v 2 ĵ [Basis Vectors î := 1, 0, ĵ := 0, 1 ] Josh Engwer (TTU) Vectors: Introduction 23 April / 30
7 3D Vector (Definition) Josh Engwer (TTU) Vectors: Introduction 23 April / 30
8 3D Vector (Representation) 3D Space: R 3 (say R-Three ) Component Form: v = v 1, v 2, v 3 [ Basis Form: v = v 1 î + v 2 ĵ + v 3 k ] Basis Vectors î := 1, 0, 0, ĵ := 0, 1, 0, k := 0, 0, 1 Josh Engwer (TTU) Vectors: Introduction 23 April / 30
9 Vectors (Translation Invariance) Two vectors are equal their components are equal they both bear the same magnitude & same direction. Merely translating a vector (as illustrated above) does not change it. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
10 Vector Algebra (Addition) Josh Engwer (TTU) Vectors: Introduction 23 April / 30
11 Vector Algebra (Addition) Josh Engwer (TTU) Vectors: Introduction 23 April / 30
12 Vector Algebra (Addition) Josh Engwer (TTU) Vectors: Introduction 23 April / 30
13 Vector Algebra (Scalar Multiplication) Josh Engwer (TTU) Vectors: Introduction 23 April / 30
14 Parallel Vectors (Definition) Definition Nonzero Vectors v, w are parallel v w v = k w for some k R. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
15 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
16 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
17 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
18 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
19 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
20 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
21 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
22 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
23 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
24 Vector Algebra (Subtraction) Find u v. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
25 Vector Algebra (Operations) Let vectors u, v R 2 and scalar k R. Then: u + v = u 1, u 2 + v 1, v 2 = u 1 + v 1, u 2 + v 2 u v = u 1, u 2 v 1, v 2 = u 1 v 1, u 2 v 2 kv = k v 1, v 2 = kv 1, kv 2 Let vectors u, v R 3 and scalar k R. Then: u + v = u 1, u 2, u 3 + v 1, v 2, v 3 = u 1 + v 1, u 2 + v 2, u 3 + v 3 u v = u 1, u 2, u 3 v 1, v 2, v 3 = u 1 v 1, u 2 v 2, u 3 v 3 kv = k v 1, v 2, v 3 = kv 1, kv 2, kv 3 Zero Vector: 0 := 0, 0 0 := 0, 0, 0 Josh Engwer (TTU) Vectors: Introduction 23 April / 30
26 Vector Algebra (Properties) Let vectors u, v, w { R 2, R 3} and scalars s, t R. Then: u + v = v + u (u + v) + w = u + (v + w) (st)u = s(tu) = t(su) u + 0 = u u u = u + ( u) = 0 (s + t)u = su + tu s(u + v) = su + sv Josh Engwer (TTU) Vectors: Introduction 23 April / 30
27 Vectors (Norm) Definition The norm of a 2D vector v = v 1, v 2 is defined to be v := v v2 2 Definition The norm of a 3D vector v = v 1, v 2, v 3 is defined to be v := v v2 2 + v2 3 Josh Engwer (TTU) Vectors: Introduction 23 April / 30
28 Vectors (Norm) Definition The norm of a 2D vector v = v 1, v 2 is defined to be v := v v2 2 Definition The norm of a 3D vector v = v 1, v 2, v 3 is defined to be v := v v2 2 + v2 3 PROOF: Use Pythagorean s Theorem. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
29 Unit Vectors & Direction Vectors Definition A unit vector v is a vector with norm one. A direction vector for vector v is defined to be v := v v Josh Engwer (TTU) Vectors: Introduction 23 April / 30
30 Fin Fin. Josh Engwer (TTU) Vectors: Introduction 23 April / 30
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