Class Notes (835,574)
United States (324,189)
Mathematics (102)
MAC 2313 (10)
Lecture 2

Lecture 2 - Dot Product.pdf

10 Pages
Unlock Document

MAC 2313
Jason Kozinski

Lecture 2 (Text Section 12.3) Dot Product Several different products can be defined for vectors. In the previous lecture we defined the product of a scalar with a vector. We now define a new product in which one vector is multiplied by a second vector. The dot product of vector u and v, represented symboli- cally as u · v, is defined by u · v = where ▯ is the angle (0 ≤ ▯ ≤ ▯) between the vectors. Note that the resulting product is a scalar. Two vectors are said to be orthogonal if their dot prod- uct is equal to zero. For nonzero vectors, this implies that the angle between the vectors is The zero vector is orthogonal to all other vectors. The definition of the dot product does not provide a conve- nient means of computation for component-wise representa- tions of vectors. This is addressed by the following Theorem: Theorem 12.1 Given two vectors u = ⟨u ;u1;u 2 a3d v = ⟨v ;v ;v1⟩,2 3 u · v = u 1 1 u v2 2u v :3 3 Example: Find the dot product of the vectors u = ⟨2;3;−5⟩ and v = ⟨0;7;4⟩. Knowing the dot product between two vectors allows for a simple computation of the angles between two vectors. From the definition of the dot product, we obtain the following re- sult: ▯ = Example: Find the angle between the vectors given in the previous example. Note if u = ⟨u 1u ;2 ⟩3 then u · u = Example: Use the dot product to calculate the length of u = ⟨−2;3;5⟩. Properties of the Dot Product Suppose u, v, and w are vectors and let c be a scalar. 1. u · v = 2. c(u · v) = 3. u · (v + w) = These properties can be readily proved using the component- wise representation of vectors. Example: Prove property 3 above. Orthogonal Projections Given two nonzero vectors u and v we can express u as the sum of two vectors, one of which is parallel to v and the other which is orthogonal to v. The component of u parallel to v is called the projection of u onto v and is represented symbolically by proj uv the length of the projection vector is r
More Less

Related notes for MAC 2313

Log In


Join OneClass

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

Sign up

Join to view


By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.