Class Notes (835,929)
United States (324,289)
Mathematics (102)
MAC 2313 (10)
Lecture 3

# Lecture 3 - Cross Product.pdf

11 Pages
64 Views

School
Department
Mathematics
Course
MAC 2313
Professor
Jason Kozinski
Semester
Summer

Description
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
More Less

Related notes for MAC 2313
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.