The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines.
(a) Use a sketch of the vector field F(x, y) = x i - y j to draw some flow lines. From your sketches, can you guess the equations of the flow lines?
(b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = -y. Then solve the differential equations to find an equation of the flow line that passes through the point (1, 1).