Q. 12 - Calculus

Question

What is the difference between the graphs of

$\mathrm{G}(x,y,z)=2\mathbf{i}-3\mathbf{j}+z\mathbf{k} and \mathrm{F}(x,y,z)=-2\mathbf{i}+3\mathbf{j}-z\mathbf{k}$

Step-by-Step Solution

Answer

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

Step 1.Given Information

What is the difference between the graphs of

$\mathrm{G}(x,y,z)=2\mathbf{i}-3\mathbf{j}+z\mathbf{k} and \mathrm{F}(x,y,z)=-2\mathbf{i}+3\mathbf{j}-z\mathbf{k}$

Step 2. The given vector is G ( x , y , z ) = 2 i − 3 j + z k   a n d   F ( x , y , z ) = − 2 i + 3 j − z k .

$$\begin{gathered} \mathrm{For} \mathrm{every} \mathrm{point} (\mathrm{x},\mathrm{y},\mathrm{z})\in {\mathrm{ℝ}}^3 \\ \mathrm{We} \mathrm{can} \mathrm{write} \mathrm{F}(x\mathit{,}y) \mathrm{as} \\ \mathrm{F}(x,y,z)=-2\mathbf{i}+3\mathbf{j}-z\mathbf{k} \\ \mathrm{F}(x,y,z)=-(2\mathbf{i}-3\mathbf{j}+z\mathbf{k}\mathbf{)} \\ \mathrm{As} \mathrm{we} \mathrm{know} \mathrm{G}(x\mathit{,}y,z)=2\mathbf{i}-3\mathbf{j}+z\mathbf{k} \\ \mathrm{So} \mathrm{we} \mathrm{can} \mathrm{write} \\ \mathrm{F}(\mathrm{x},\mathrm{y},z)=-\mathrm{G}(x\mathit{,}y,z) \\ \mathrm{Hence} \mathrm{F}(\mathrm{x},\mathrm{y},z)=-\mathrm{G}(x\mathit{,}y,z); \mathrm{at} (0,0) \end{gathered}$$