Vector Fields

 Vector Fields

Definition   A vector field is a function F that assign to point whether it's in two dimension or three dimension 

the best way to understand the vector fields is by drawing it

and we have a special notations for vector fields 

F(x,y,z)=Q(x,y,z) i+ P(x,y,z) j+ R(x,y,z) k

or

F(x,y,z)=<Q(x,y,z), P(x,y,z), R(x,y,z)>

For Example:\[( - 1,1) \to \,\,\, < 1, - 1 > \]

this directional line will start at point (-1,1) and it will move 1 unit to write and 1 unit down

                                             

Important notes to remember:

Position vector is a vector from the origin to a point, so it’s like a radius of the circle that centered at the origin.

If the dot product equal zero then two vectors is perpendicular to each other.

Example 2    A vector field on R is defined by . Describe F by sketching some of the vectors F(x,y)

Solution: 

the lines appears like tangents of a circles

what is the dot product of our vector field with the position vector?

this shows they are perpendicular to each other therefore tangent to a circle with center the origin