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Mathematics (102)
MAC 2313 (10)
Lecture 3

Lecture 3 - Cross Product.pdf

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MAC 2313
Jason Kozinski

Lecture 3 (Text Section 12.4) Cross Product In this lecture we again define a new product for vectors. In this case, the multiplication of two vectors yields a third vector with some very interesting properties and applications. The cross product of the vectors u and v, represented symbolically as u × v, is defined to be a vector of magnitude |u × v| = where ▯ is the angle (0 ≤ ▯ ≤ ▯) between the vectors and where the resulting product vector has a direction given by the right hand rule. According to the right hand rule, you point the out- stretched fingers of the right hand in the direction of the vec- tor u and curl these fingers in the direction of v (in the same sense as the angle ▯); the direction in which your thumb is pointing during this operation is the direction of u × v. Example: According to this rule, if your paper corresponds to the x;y-plane with the positive y-axis pointing to the top of the page and the positive x-axis pointing to the right of the page then, i×j is perpendicular to the page pointing upward and j × i is perpendicular to the page pointing downward. The first results of the definition of the cross products are given in the following theorem: Theorem 12.3. 3 Let u and v be two vectors in R . 1. The vectors u and v are parallel (▯ = 0 or ▯ = ▯) if and only if 2. If u and v are two sides of a parallelogram then the area of the parallelogram is The definition of the cross product does not provide a con- venient means of computation for component-wise represen- tations of vectors. This is addressed by the following Theorem: Theorem 12.6 Let u = ⟨u ;u ;1 ⟩ 2nd v3= ⟨v ;v ;v ⟩, then1 2 3 ▯ ▯ ▯ i j k ▯ u × v = ▯u u u : ▯ ▯ 1 2 3 ▯ ▯v 1 v 2 v 3 ▯ The determinant in the theorem can be computed in many ways; perhaps the simplest technique is to write the three columns of the matrix and add two additional columns by re- peating the first and second columns. Next proceed from left t
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