General Vectors, the Dot Product, & Planes.pdf - Chapter 1 Introduction to multivariable functions and vectors 1.3 General vectors dot products and

General Vectors, the Dot Product, & Planes.pdf -...

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Chapter 1 Introduction to multivariable functions and vectors 1.3 General vectors, dot products and planes R , R 2 and R 3 are nice because they can be visualized, but it is still natural to consider vectors in higher dimensions. Indeed, some mathematical models involve more than 3 variables. For completeness we provide the following: Definition 1.3.1 (General vectors, addition, scalar multiplication, magni- tude) . A vector in R n is an array of n number ~v = h v 1 , . . . , v n i in angled brackets. Given another vector ~w = h w 1 , . . . , w n i and a scalar α R we have the following addition and scalar multiplication formulas: ~v + ~w = h v 1 + w 1 , . . . , v n + w n i (1.3.1) α · ~v = h αv 1 , . . . , αv n i . (1.3.2) Although higher dimension geometry seems far-fetched we also provide the following: Definition 1.3.2. Given points P = ( p 1 , . . . , p n ) , Q = ( q 1 , . . . , q n ) R n the displacement vector from P to Q is ~ PQ = h q 1 - p 1 , . . . , q n - p n i = h Q - P i . (1.3.3) The magnitude of a vector ~v is given by the Pythagorean formula: | ~v | = q v 2 1 + . . . + v 2 n . (1.3.4) 1
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CHAPTER 1. MULTIVARIABLE FUNCTIONS; VECTORS 2 And the distance between two points is by the formula: d ( P, Q ) = ~ PQ . (1.3.5) The remarkable dot product Geometry is all about angles and distances. The dot product completely defines what we know as the Euclidean geometry on R n . Because we consider vectors as arrows that are free to float around R n it makes sense to try to give a precise definition of the angle between two vectors. Definition 1.3.3. Let ~v and ~w be vectors in R n then the angle between ~v and ~w is the angle θ shown below with the prescribed side lengths | ~w | | ~v - ~w | | ~v | θ Which can be calculated using the Law of cosines to be θ such that | ~v - ~w | 2 = | ~w | 2 + | ~v | 2 - 2 | ~v || ~w | cos( θ ) (1.3.6) In particular if you can calculate all those magnitudes you can solve for cos( θ ) and then arccos it. Informally it’s the angle between the vectors when you put their backsides together. Doing this in 2D is straightforward. Example 1.3.1. What is the angle between the vectors ~w = h- 5 , 4 i and ~v = h 2 , 2 i ? x y ~w h- 5 , 4 i θ ~u h 2 , 2 i
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CHAPTER 1. MULTIVARIABLE FUNCTIONS; VECTORS 3 Previously, by normalizing and using arccos we found the angle θ to be about 141 . The angle between h 2 , 2 i originating at the origin (0 , 0) and the x -axis is 45 ; so the angle between the two vectors is about 141 - 45 = 96 . In R 3 the angle between two vectors can be found similarly by putting them end-to-end and measuring the angle in the plane that contains them both. This is really hard to do, and it’s why the dot product is great. Definition 1.3.4 (Dot product (algebraic)) . Let ~v = h v 1 , . . . , v n i and ~w = h w 1 , . . . , w n i be two vectors, then their dot product is the following scalar . ~v · ~w = v 1 w 1 + . . . + v 1 w n . (1.3.7) Although this operation looks silly, its importance cannot be overstated.
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