Vector Fields — Question 1

Question 1

Consider the planar vector field $$\mathbf F(x,y)=\langle x,-y\rangle.$$

Tasks

1. Evaluate $\mathbf F$ at $(1,2)$, $(-2,1)$, and $(0,0)$.

2. Find the magnitude and direction of $\mathbf F(1,2)$.

3. Describe the field on the coordinate axes and identify every zero vector.

Question 1 – Solution

Strategy. Evaluate the two components pointwise, then use the component signs to understand the field globally.

Step 1: Sample vectors $$\mathbf F(1,2)=\langle 1,-2\rangle,\qquad \mathbf F(-2,1)=\langle-2,-1\rangle,\qquad \mathbf F(0,0)=\langle 0,0\rangle.$$

Step 2: Magnitude and direction At $(1,2)$, $$|\mathbf F|=\sqrt{1^2+(-2)^2}=\boxed{\sqrt 5}, \qquad \widehat{\mathbf F}=\boxed{\left\langle\frac 1{\sqrt 5},-\frac 2{\sqrt 5}\right\rangle}.$$

Step 3: Axis behavior and zeros On the $x$-axis, $\mathbf F(x,0)=\langle x,0\rangle$, so arrows point away from the origin. On the $y$-axis, $\mathbf F(0,y)=\langle 0,-y\rangle$, so arrows point toward the origin. Finally, $$\mathbf F(x,y)=\mathbf 0\iff x=0\ \text{and}\ y=0.$$ Thus the origin is the field’s .

Verification The plotted arrows reverse horizontally when $x$ changes sign and vertically when $y$ changes sign, exactly matching the formula.