lec22 - 22 22.1 22.1.1 APPENDIX 1: MATH FACTS Vectors...

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22 APPENDIX 1: MATH FACTS 22.1 Vectors 22.1.1 Defnition A vector has a dual defnition: It is a segment oF a a line with direction, or it consists oF its projection on a reFerence system 0 xyz , usually orthogonal and right handed. The frst Form is independent oF any reFerence system, whereas the second (in terms oF its components) depends directly on the coordinate system. Here we use the second notation, i.e., x is meant as a column vector, whose components are Found as projections oF an (invariant) directed segment on a specifc reFerence system. We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction . ±or example, in three-space, we write a vector in terms oF its components with respect to a reFerence system as 2 1 γa = . 7 The elements oF a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. z 2 x y 1 7 a
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22.1 Vectors 111 1. Vector addition: γa + γ b = γ c 2 3 5 1 + 3 = 4 . 7 2 9 Graphically, addition is stringing the vectors together head to tail. 2. Scalar multiplication: 2 4 2 × 1 = 2 . 7 14 22.1.2 Vector Magnitude The total length of a vector of dimension m , its Euclidean norm, is given by m ± 2 || γx || = x i . i =1 This scalar is commonly used to normalize a vector to length one. 22.1.3 Vector Dot or Inner Product The dot product of two vectors is a scalar equal to the sum of the products of the corre- sponding components: m x γ y = γ y = T γ · x i y i . i =1 The dot product also satisFes γ · y = y || cos χ, || |||| γ where χ is the angle between the vectors.
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± 112 22 APPENDIX 1: MATH FACTS 22.1.4 Vector Cross Product x and γy is another vector γ z , γ y The cross product of two three-dimensional vectors γ x × γ = γ z , whose 1. direction is normal to the plane formed by the other vectors, 2. direction is given by the right-hand rule, rotating from γx to γ y , 3. magnitude is the area of the parallelogram formed by the vectors the cross product of parallel vectors is zero and 4. (signed) magnitude is equal to || γ y || sin χ , where χ is the angle between the x |||| γ vectors, measured from to γ y . In terms of their components, × γ y = ˆ ˆ ˆ i j k x 1 x 2 x 3 y 1 y 2 y 3 = ( x 2 y 3 x 3 y 2 ) ˆ i j ( x 1 y 2 x 2 y 1 ) ˆ ( x 3 y 1 x 1 y 3 ) ˆ k . 22.2 Matrices 22.2.1 Defnition A matrix, or array, is equivalent to a set of column vectors of the same dimension, arranged side by side, say 2 3 A = [ γa γ b ] = 1 3 . 7 2 This matrix has three rows ( m = 3) and columns ( n = 2); a vector is a special case of a matrix with one column. Matrices, like vectors, permit addition and scalar multiplication.
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lec22 - 22 22.1 22.1.1 APPENDIX 1: MATH FACTS Vectors...

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