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Fun with Vectors

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Definition A vector is a quantity that has both magnitude and direction Examples?

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Represented by an arrow A (initial point) B (terminal point) v, v, or AB

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If two vectors, u and v, have the same length and direction, we say they are equivalent u v

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a b Vector addition

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a b Vector addition: a + b

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a b a+ba+b

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a b a+ba+b

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Scalar Multiplication a

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a 2a2a -a -½ a

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Subtraction b a

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b a -b

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Subtraction b a -b a+(-b)

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Subtraction b a -b a+(-b)

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Subtraction b a -b a+(-b) If a and b share the same initial point, the vector a-b is the vector from the terminal point of b to the terminal point of a

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Let’s put these on a coordinate system We can describe a vector by putting its initial point at the origin. We denote this as a= where (a 1,a 2 ) represent the terminal point

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Graphically x y a= (a 1,a 2 ) y x z v= a b c (a,b,c)

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Given two points A=(x1,y1) and B=(x2,y2), The vector v = AB is given by v = …or in 3-space, v =

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Graphically A=(-1,2) B=(2,3) A B v = = v

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Recall, a vector has direction and length Definition: The magnitude of a vector v = is given by

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Properties of Vectors Suppose a, band c are vectors, c and d are scalars 1.a+b=b+a 2.a+(b+c)=(a+b)+c 3.a+0=a 4.a+(-a)=0 5.c(a+b)=ca+cb 6.(c+d)a=ca+da 7.(cd)a=c(da) 8.1a=a

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Standard Basis Vectors Definition: vectors with length 1 are called unit vectors

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Example: We can express vectors in terms of this basis a = a = 2i -4j+6k Q. How do we find a unit vector in the same direction as a? A. Scale a by its magnitude

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Example a =

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12.3 The Dot Product Motivation: Work = Force* Distance Box F D Fx Fy

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Box D F Fx Fy To find the work done in moving the box, we want the part of F in the direction of the distance

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One interpretation of the dot product Where is the angle between F and D

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A more useful definition You can show these two definitions are equal by considering the following triangle and applying the law of cosines! See page 808 for details y x z b a-b a Think, what is |a| 2 ?

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Example a=, b= Find a. b and the angle between a and b

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The Dot Product If a = and b= then The dot product of a and b is a NUMB3R given by

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The Dot Product a and b are orthogonal if and only if the dot product of a and b is 0 Other Remarks: a b

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Properties of the dot product Suppose a, b, and c are vectors and c is a scalar 1.a. a=|a| 2 2.a. b=b. a 3.a. (b+c) = (a. b)+(a. c) 4.(ca). b=c(a. b)=a. (cb) 5.0. a=0

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Yet another use of the dot product: Projections a. b=|a| |b| cos( ) Think of our work example: this is ‘how much’ of b is in the direction of a b a |b| cos( )

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We call this quantity the scalar projection of b on a Think of it this way: The scalar projection is the length of the shadow of b cast upon a by a light directly above a

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Q. How do we get the vector in the direction of a with length comp a b? A.We need to multiply the unit vector in the direction of a by comp a b. We call this the vector projection of b onto a

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Examples/Practice!

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Key Points Vector algebra: addition, subtraction, scalar multiplication Geometric interpretation Unit vectors The dot product and the angle between vectors Projections (algebraic and geometric)

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Vectors and the Geometry of Space

Vectors and the Geometry of Space

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