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Lecture 2

Lecture 2 - Dot Product.pdf

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Department
Mathematics
Course
MAC 2313
Professor
Jason Kozinski
Semester
Summer

Description
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
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