Q. 11 - Calculus

Question

What is the difference between the graphs of

$\mathrm{G}(x,y)=x\mathbf{i}+\mathrm{y}\mathbf{j} and \mathrm{F}(x,y)=-x\mathbf{i}+(-y)\mathbf{j}$

Step-by-Step Solution

Answer

The difference between the graphs are $\mathrm{F}(\mathrm{x},\mathrm{y})=-\mathrm{G}(x\mathit{,}y); \mathrm{at} (0,0)$.

Step 1.Given Information

What is the difference between the graphs of

$\mathrm{G}(x,y)=x\mathbf{i}+\mathrm{y}\mathbf{j} and \mathrm{F}(x,y)=-x\mathbf{i}+(-y)\mathbf{j}$

Step 2. The given vector is G ( x , y ) = x i + y j   a n d   F ( x , y ) = − x i + ( - y ) j .

$$\begin{gathered} \mathrm{For} \mathrm{every} \mathrm{point} (\mathrm{x},\mathrm{y})\in {\mathrm{ℝ}}^2 \\ \mathrm{We} \mathrm{can} \mathrm{write} \mathrm{F}(x\mathit{,}y) \mathrm{as} \\ \mathrm{F}(x,y)=-x\mathbf{i}+(-y)\mathbf{j} \\ \mathrm{F}(x,y)=-x\mathbf{i}-y\mathbf{j} \\ \mathrm{F}(x,y)=-(x\mathbf{i}+y\mathbf{j}\mathbf{)} \\ \mathrm{As} \mathrm{we} \mathrm{know} \mathrm{G}(x\mathit{,}y)=x\mathbf{i}+\mathrm{y}\mathbf{j} \\ \mathrm{So} \mathrm{we} \mathrm{can} \mathrm{write} \\ \mathrm{F}(\mathrm{x},\mathrm{y})=-\mathrm{G}(x\mathit{,}y) \\ \mathrm{Hence} \mathrm{F}(\mathrm{x},\mathrm{y})=-\mathrm{G}(x\mathit{,}y); \mathrm{at} (0,0) \end{gathered}$$