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Q. 12

Expert-verified
Found in: Page 1095

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# What is the difference between the graphs of$\mathrm{G}\left(x,y,z\right)=2\mathbf{i}-3\mathbf{j}+z\mathbf{k}and\mathrm{F}\left(x,y,z\right)=-2\mathbf{i}+3\mathbf{j}-z\mathbf{k}$

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

See the step by step solution

## Step 1.Given Information

What is the difference between the graphs of

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

## Step 2. The given vector is G(x,y,z)=2i−3j+zk and F(x,y,z)=−2i+3j−zk.

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