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Pure Vector

2.0 Vectors in different dimensions


2.1    1-Dimension


2.2    2-Dimension

2.3    3-Dimension (3D)






As you might have noticed, there are different ways to position the axes. However, there is a convention to draw the axes. The following rules can be used in deciding on the directions of the axes.

2.3.1 The cork-screw rule



                                                                        
                                                                          Fig 2.4

                                        
                                                                                         


                                                
                                                                           Fig 2.5





The cork will move up when turned in an anticlockwise direction. Similarly, starting from x-direction and moving in the y-direction (anticlockwise), the z - direction will be the direction of the cork, that is upward.

                                                   
                                                                                         


                             
On the contrary, in the following example, the z axis will point downwards.


                                   





                                                                         


                                                 Fig 2.6



2.3.2 The Right Hand Grip Rule




                                   
                                                                                          Fig 2.7


                        


Taking the grip from x to y (that is anticlockwise direction), the thumb gives the direction of the z axis.



                                   
                                                                                          Fig 2.8


                                   




3.0    3-D Coordinates











NOTE : A right pyramid is one whose vertex is vertically above the centre of its base.






4.0   3D Vectors


4.1 Distance between 2 points




4.2 Midpoint



4.3 Segment Formula




4.5 Base vectors


4.6 Position Vectors






4.7 Modulus or Magnitude of a vector






4.8 Unit vector


4.10 Dot Product (Scalar Product)




4.11 Angle between two vectors









4.13 Perpendicular vectors





4.14 Parallel Vectors






4.15 Collinear Points



4.16 Geogebra links


4.17 Miscellaneous Example