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Lecture

# lect136_4_w14.pdf

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School
University of Waterloo
Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

Description
Monday, January 13 − Lecture 4 : Dot products and norms Concepts: n 1. dot product of two vectors in ℝ . 2. algebraic properties of dot product of two vectors. 3. length or norm of a vector in ℝ . n 4. unit vector in the direction of a n 5. angle between two vectors in ℝ . 6. distance between two points in ℝ . n 7. Cauchy-Schwartz inequality n 8. orthogonal vectors or ortnogonal sets in ℝ . 9. orthonormal sets in ℝ . 4.1 Dot product operation on ℝ definition. Operation What we start with : What we get : n 1 number x⋅y in ℝ : 2 vectors x and y in ℝ : x⋅y = x y + x y + ..., x y . x = (x 1 x2, ...,nx ) 1 1 2 2 n n Dot product and Other notation for dot-product: y = ( y , y , ...,y ). x⋅y = < x, y > 1 2 n 4.1.1 Examples − If x = (2, 1, 7) and y = (3, − 9, 0) verify that x ⋅ y = −3. n 4.1.2 Theorem − Algebraic properties of dot product. Suppose x, y and z are in ℝ and α in ℝ. Then DP1 x ⋅x ≥ 0 DP2 x ⋅y = y ⋅x DP3 (αx) ⋅y = α(x ⋅y) DP4 x ⋅y + z) = x ⋅y + x ⋅z Proof is straight forward and is left as an exercise. Norm or length of a vector 4.2 Definition − For a vector x in ℝ , the length or norm of x, denoted by || x || , is the number n 4.2.1 Properties of the norm in ℝ . 3 It can be shown that the norm on ℝ satisfies 3 properties: N1 For every vector x, || x || ≥ 0. Furthermore, ||x|| = 0 ⇔ x = 0. (Positive definite property) N2 || αx || = |α| || x || for all scalars α, and x in V. (Homogeneous property) N3 || x + y || ≤ || x || + || y || for all x, y in V(Triangle inequality property) n 4.2.2 Observation − If a is any non-zero vector in ℝ , then the vector defined as b = (1/|| a ||)a has norm equal to 1 since The vector a/|| a || is called the unit vector in the direction of a. Note that if b is a vector which points in the same direction as a then This is true since if b and a point in the same direction then they both have the same unit vector, i.e., a / || a || = b / || b ||. 4.2.3 Remark – In ℝ the norm, thus defined, leads to the familiar distance formula between two points x and y. The distance betweenx = (x , x ) and y = (y , y ) is 1 2 1 2 n Definition − Let a and b be two vectors in ℝ . The distance between a and b is defined as || b − a || n 4.3 Theorem − Cauchy-Schwartz inequality. For any vectors x and y in ℝ | x⋅y | ≤ || x || || y || Proof : n Let x, y be two vectors in ℝ and t be a variable scalar. For any real number t: (Be sure to verify the equality between the expression in the f
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