Vectors

 

 

Physical quantities usually fall into one of two categories:-

 

 

·                   Scalar

·                   Vector

 

 

A scalar quantity is defined totally by its magnitude (its size) and units.

 

A few scalars (not exhaustive)

Scalar       Examples	magnitude	units
Time               20 mins	20	minutes
Mass                 30kg	30	kg
Volume       1.77889 m3	1.77889	m3
Area            66.999  km2	66.999	km2

 

 

 

 

A few vectors (not exhaustive)

Vector       Examples	magnitude	units	direction
Acceleration   due to gravity	9.8	m/s2	Vertically down
Wind     20 knots NE	20	knots	From North East
Velocity    10 mph due south	10	mph	Due south
Weight of  mass of 30kg    (weight = mass x gravity)	294	N	Vertically down

 

 

·        Writing vectors

 

A vector  can be drawn as a line, with the length of the line representing the magnitude

 and the direction indicated by an arrowhead.

 

 

 

 

 

 

 

 

 

The vectors below are shown in component form.

 

 

                  

 

 

 

 

 

 

           

 

 

 

          

 

Example

 

 

 

    

 

                                                        

             

·        Equal Vectors

 

 

 

 

 

Direction is important

 

 

                                                                     

 

 

 

 

 

 

 

 

·        Magnitude

 

      

 

 

 

 

Example

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example

 

 

 

 

 

·       Addition of Vectors

 

      

 

 

 

  

 

 

        

 

 

 

 

   Example

 

 

        

 

·        Subtraction of vectors

 

           

 

 

 

 

 

 

 

 

 

·        The Zero Vector

 

 

 

 

 

 

 

 

·        Unit Vector

 

 

                    

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·        Position Vectors

 

 

 

 

                   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 D Vectors

 

 

 

 

 

 

 

 

 

 

·        Multiplication of a vector by a Scalar

 

 

 

 

 

 

       

 

 

 

·        Collinearity

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·        Section Formula

   

     

 

 

            

 

                           

 

 

 

 

 

      Example

 

          

 

 

 

·        The Scalar Product : Dot Product

 

        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·        Component Form of Dot Product

 

                    

 

                Examples

 

             

 

 

·        Angle between vectors