Question

Short answer

Step-by-Step Solution

Step-by-step solution

Concept

To sketch a vector field, draw the arrow representing the vector\({\rm{F}}\left( {{\rm{x,y}}} \right){\rm{ = - }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{i + }}\left( {{\rm{y - x}}} \right){\rm{j}}\)starting at point\(\left( {{\rm{x, y}}} \right)\).

It is impossible to do this for all points\(\left( {{\rm{x, y}}} \right)\)but we can gain a reasonable impression of\({\rm{F}}\)by doing it for a few representative points in the domain.

Given Information.

The given vector field is \(F\left( {x,y} \right) = - \frac{1}{2}i + \left( {y - x} \right)j\).

Calculation.

We will first calculate several representative values of \(F\left( {x,y} \right)\) in the table form, and then, by using the concept, draw the corresponding vectors to represent the vector field.

Several representative values of \(F\left( {x,y} \right)\) are:

\(\left( {x,y} \right)\)	\(F\left( {x,y} \right)\)
\((1,0)\)	\(\left( { - \frac{1}{2}, - 1} \right)\)
\((2,2)\)	\(\left( { - \frac{1}{2},0} \right)\)
\((0,1)\)	\(\left( { - \frac{1}{2},1} \right)\)
\(( - 2,2)\)	\(\left( { - \frac{1}{2},4} \right)\)
\(( - 1,0)\)	\(\left( { - \frac{1}{2},1} \right)\)
\(( - 2, - 2)\)	\(\left( { - \frac{1}{2},0} \right)\)
\((0, - 1)\)	\(\left( { - \frac{1}{2}, - 1} \right)\)
\((2, - 2)\)	\(\left( { - \frac{1}{2}, - 4} \right)\)

Sketch the vector field.

The sketch of the vector field will be: