12.1 – The Three Dimensional Coordinate System•The distance formula o=-+-+(-)P1P2x2 x1y2 y1z2 z1Where =(,,)P1x1 y1 z1and =(,,)P2x2 y2 z2•The Equation of a Sphereo(-)+(-)+(- )=x h 2y k 2z l 2r2oWhere (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, o12.2-Vectors•Know how to add and subtract vectors geometrically and algebraically •The length of a vector (magnitude) can be determined byo=++aa12a22a32, where =<,,>aa1 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, #31o12.3-Dot Product•The dot product of two vectors areo<,,> <,,> =++a1 a2 a3dotb1 b2 b3a1b1a2b2a3b3•Review the properties of the dot products on page 807•The geometric form of the dot product iso=a dot ba|b|cosθ•If the dot product of two vectors is zero then they are perpendicular, think cosθ•Review 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 vectoroooooo•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,o12.4-Cross Product•The cross product of two vectors <,,><,,> a1 a2 a3 and b1 b2 b3is o=<-, -, ->a x ba2b3 a3b2a3b1 a1b3a1b2 a2b1o: =Equivalentlya x bijka1a2a3b1b2b3•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

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 sinθ•Review the properties of cross products on pg 818•Torque is calculated by the cross product of distance and force. At a 90 degree angle the torque will be the greatest, think sinθ•The 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 Planes•In 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 line•From this we can obtain the Vector form of the line, we can obtain the parametric from and symmetric from of a line.