Review test 1 - 12.1 The Three Dimensional Coordinate...

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12.1 – The Three Dimensional Coordinate System The distance formula o = - + - +( - ) P1P2 x2 x1 y2 y1 z2 z1 Where =( , , ) P1 x1 y1 z1 and =( , , ) P2 x2 y2 z2 The Equation of a Sphere o ( - ) +( - ) +( - ) = x h 2 y k 2 z l 2 r2 o Where (h,k,l) is the center of the sphere and r is the radius Extra Exercises: Pg 797 #5, #7, #10, #13, #17, #25, #29, o 12.2-Vectors Know how to add and subtract vectors geometrically and algebraically The length of a vector (magnitude) can be determined by o = + a a12 a22 a32 , where =< , > a1 a2 a3 Recall the problems of application of vectors using components The standard basis vectors are <I,j,k> Extra Exercises: pg 805 #3,#9, #19, #29, #31 o 12.3-Dot Product The dot product of two vectors are o < , , > < , , > = + + a1 a2 a3 dot b1 b2 b3 a1b1 a2b2 a3b3 Review the properties of the dot products on page 807 The geometric form of the dot product is o = a dot b a|b|cosθ If the dot product of two vectors is zero then they are perpendicular, think cos θ Review the difference between scalar projections and vector projections o o Scalar o Vector
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Projection Projection o Form ula o = compab a dot ba o = projab a dot baa|a| o Form o A Scalar o A vector (unit vector) o Speci al Properties o Draw a picture and see that o Cosθ is the vector adjacent is the vector that we want o Look at pg 810 o This is a Unit Vector in the direction of the original vector o o o o o o Work =F dot D, deals with dot products. REMEMBER: *--* the force and distance must be tail to tail for the angle to be correct *--* Extra Exercises: Pg 812 #7, # 9, #21, #35, o 12.4-Cross Product The cross product of two vectors < , , > < , , > a1 a2 a3 and b1 b2 b3 is o =< - , - , - > a x b a2b3 a3b2 a3b1 a1b3 a1b2 a2b1 o : = Equivalently a x b ijka1a2a3b1b2b3 The vector (a x b) is perpendicular to both a and b It is your thumb when you point your fingers in the direction of a and b, using your right hand (right hand rule). The cross products geometric interpretation is
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o = a x b |a ||b|sinθ o Notice that this is the magnitude of the cross product, it is a scalar. The magnitude of a x b is equal to the area of the parallelogram of a and b.
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This note was uploaded on 04/19/2008 for the course MATH 123 taught by Professor Cox during the Spring '08 term at SUNY Fredonia.

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Review test 1 - 12.1 The Three Dimensional Coordinate...

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