Review test 1 - 12.1 The Three Dimensional Coordinate System The distance formula o Where and The Equation of a Sphere o Where(h,k,l is the center of

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

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12.1 – The Three Dimensional Coordinate SystemThe distance formula o=-+-+(-)P1P2x2 x1y2 y1z2 z1Where =(,,)P1x1 y1 z1and =(,,)P2x2 y2 z2The Equation of a Sphereo(-)+(-)+(- )=x h 2y k 2z l 2r2oWhere (h,k,l) is the center of the sphere and r is the radiusExtra Exercises: Pg 797 #5, #7, #10, #13, #17, #25, #29, o12.2-VectorsKnow how to add and subtract vectors geometrically and algebraically The length of a vector (magnitude) can be determined byo=++aa12a22a32, where =<,,>aa1 a2 a3Recall the problems of application of vectors using componentsThe standard basis vectors are <I,j,k>Extra Exercises: pg 805 #3,#9, #19, #29, #31o12.3-Dot ProductThe dot product of two vectors areo<,,> <,,> =++a1 a2 a3dotb1 b2 b3a1b1a2b2a3b3Review the properties of the dot products on page 807The geometric form of the dot product iso=a dot ba|b|cosθIf the dot product of two vectors is zero then they are perpendicular, think cosθReview the difference between scalar projections and vector projectionsooScalar oVector
ProjectionProjectionoFormulao=compabadot bao=projabadot baa|a|oFormoA ScalaroA vector (unit vector)oSpecial PropertiesoDraw a picture and see thatoCosθ is the vector adjacent is the vector that we wantoLook at pg 810oThis is a UnitVectorin the direction of the original vectorooooooWork =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,o12.4-Cross ProductThe cross product of two vectors <,,><,,> a1 a2 a3 and b1 b2 b3is o=<-, -, ->a x ba2b3 a3b2a3b1 a1b3a1b2 a2b1o: =Equivalentlya x bijka1a2a3b1b2b3The vector (a x b) is perpendicular to both a and bIt 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
o=a x b|a ||b|sinθoNotice 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.If two vectors are parallel then their cross product is zero, think of sinθReview the properties of cross products on pg 818Torque is calculated by the cross product of distance and force. At a 90 degree angle the torque will be the greatest, think sinθThe volume of a parallelepiped is found byo== Va dot b x ca1a2a3b1b2b3c1c2c3oNotice this is a scalar, not a vector.Extra Exercises: Pg 820 #5, #27, #31, #37o12.5-Equations of Lines and PlanesIn this section, it is absolutely pivotal that you draw pictures. I can’t draw pictures that well on Microsoft Word, so you will have to ask me if you have trouble seeing something.The standard form for parameterizing a line is thato=+, rr0tvowhere v is a vector on the line and ro is some point onthe lineFrom this we can obtain the Vector form of the line, we can obtain the parametric from and symmetric from of a line.