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Question

# 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.Use a sketch of the vector field $\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}$ to draw some flow lines. From your sketches, can you guess the equations of the flow lines?

Solution

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If we take the starting point $\left(1, 0\right)$, then

$F\left(1, \ 0\right)=i.$

Next, we draw the vector $\left\langle 1, \ 0 \right\rangle$ starting at the point $\left(1, 0\right)$ as it is shown in Figure 1.
If we take the starting point $\left(0, 1\right)$, then

$F\left(0, \ 1\right)=-j.$

Next, we draw the vector $\left\langle 0, \ -1\right\rangle$ starting at the point $\left(0, 1\right)$ as it is shown in the Figure 1.

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