Question

match each vector field with its graph.

1. $mathbf{f}(x,y)=langle -y,x

angle$

1. $mathbf{f}(x,y))=langle x,-y

angle$

1. $mathbf{f}(x,y))=langle x,y

angle$

1. $mathbf{f}(x,y))=langle y,x

angle$
(click on a graph to enlarge it)
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Explanation:

Step1: Analyze \( \mathbf{F}(x,y)=\langle -y,x

angle \)
At the point \((x,0)\), the vector is \(\langle0,x
angle\) (vertical). At \((0,y)\), the vector is \(\langle -y,0
angle\) (horizontal). This is a rotational vector field.

Step2: Analyze \( \mathbf{F}(x,y)=\langle x,-y

angle \)
At the point \((x,0)\), the vector is \(\langle x,0
angle\) (horizontal). At \((0,y)\), the vector is \(\langle0,-y
angle\) (vertical). Vectors point towards the fourth - first quadrant direction for positive \(x\) and negative \(y\) and vice - versa.

Step3: Analyze \( \mathbf{F}(x,y)=\langle x,y

angle \)
At the point \((x,0)\), the vector is \(\langle x,0
angle\) (horizontal). At \((0,y)\), the vector is \(\langle0,y
angle\) (vertical). Vectors point away from the origin.

Step4: Analyze \( \mathbf{F}(x,y)=\langle y,x

angle \)
At the point \((x,0)\), the vector is \(\langle0,x
angle\) (vertical). At \((0,y)\), the vector is \(\langle y,0
angle\) (horizontal).

Answer:

1. \( \mathbf{F}(x,y)=\langle -y,x

angle\) matches graph \(C\)

1. \( \mathbf{F}(x,y)=\langle x,-y

angle\) matches graph \(B\)

1. \( \mathbf{F}(x,y)=\langle x,y

angle\) matches graph \(D\)

1. \( \mathbf{F}(x,y)=\langle y,x

angle\) matches graph \(A\)