MIT2_017JF09_ch13

# MIT2_017JF09_ch13 - 13 MATH FACTS 13 101 MATH FACTS 13.1...

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± ² ³ 13 MATH FACTS 101 13 13.1 Vectors 13.1.1 Deﬁnition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction . For example, in three-space, we write a vector in terms of its components with respect to a reference system as 2 ±a = 1 . 7 The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 1. Vector addition: + ± b = ± 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 13.1.2 Vector Magnitude The total length of a vector of dimension m , its Euclidean norm, is given by ² m || ±x || = ´ x 2 i . i =1 This scalar is commonly used to normalize a vector to length one.

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± ² ² ² ² 13 MATH FACTS 102 13.1.3 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 · ± y = T ± y = x i y i . i =1 The dot product also satisﬁes · ± y = || |||| ± y || cos θ, where θ is the angle between the vectors. 13.1.4 Vector Cross Product The cross product of two three-dimensional vectors and ± y is another vector ± z , × ± y = ± z , whose 1. direction is normal to the plane formed by the other two vectors, 2. direction is given by the right-hand rule, rotating from to ± y , 3. magnitude is the area of the parallelogram formed by the two vectors the cross product of two parallel vectors is zero and 4. (signed) magnitude is equal to || ±y || sin θ , where θ is the angle between the two x |||| ± vectors, measured from to ± y . In terms of their components, ² i j ˆ ² i ˆ ˆ k ( x 2 y 3 x 3 y 2 ) ˆ ² ² × ± y = ² x 1 x 2 x 3 ² = ( x 3 y 1 x 1 y 3 ) ˆ j . ² ² ² y 1 y 2 y 3 ² ( x 1 y 2 x 2 y 1 ) k ˆ 13.2 Matrices 13.2.1 Deﬁnition A matrix, or array, is equivalent to a set of column vectors of the same dimension, arranged side by side, say 23 A =[ ±a ± b ]= 13 .
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MIT2_017JF09_ch13 - 13 MATH FACTS 13 101 MATH FACTS 13.1...

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