I need help to sketch the vector field F(x,y) = 2xi - yj by choosing four relevant vectors in each quadrant. Then I need to find and sketch the equation of the field line that passes through the...

I need help to sketch the vector field F(x,y) = 2xi - yj by choosing four relevant vectors in each quadrant. Then I need to find and sketch the equation of the field line that passes through the point (x,y) = (2,2). It is possible to choose vectors where x = 0 and y = 0.

Could you please help?

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

For each `x,y` the point `(x, y)` is the point where a vector starts and F(x, y) is a pair of its components. For `x=y=0` we have `F(x,y)=vec0` and we cannot draw it with an arrow because it has no direction.

For the given point `(2,2)` its field vector is `F(2,2) = (4, -2),` it is the direction vector of the corresponding line. This means the slope of this line is `y/x = -2/4 = -1/2` and the equation is  `y-2=-1/2 (x-2),` or `y=-1/2 x+3.`

To draw relevant vectors, choose some integer arguments of `F(x,y).` It is simple to start a vector `F(x,y)` at the point `(x,y):` we just need add `(x,y)` and `(2x, -y) ` and obtain `(3x, 0).` This way all vectors point to the x-axis.

The attached picture uses the starting points (1,1), (1,2), (1,3), (1,4) for the first quadrant, (-1,1), (-2,2), (-3,3), (-4,4) for the second, (-1,-1), (-2,-1), (-1,-2), (-2,-2) for the third and (1,-1), (2,-1), (3,-1), (4,-1) for the fourth. Actually, the field is symmetric across x and y axes.

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)