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Question
Find the vector equations of the coordinate planes.
Solution
\[ \text{ Vector equation of XY-plane}\]
\[\text{ This plane is passing through the origin whose position vector is } \vec{a} = 0^\to \text{ and perpendicular to z-axis whose position vector is } \hat{k} .\]
\[\text{ So, the equation of the XY-plane is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \hat{k} = \vec{0} . \hat{k} \]
\[ \Rightarrow \vec{r} . \hat{k} = 0\]
\[\text{ Vector equation of YZ-plane } \]
\[\text{ This plane is passing through the origin whose position vector is } a^\to = 0^\to \text{ and perpendicular tox-axis whose position vector is } \hat{i} .\]
\[\text{ So, the equation of the YZ-plane is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \hat{i} = \vec{0} . \hat{i} \]
\[ \Rightarrow \vec{r} . \hat{i} = 0\]
\[\text{ Vector equation of XZ-plane } \]
\[ \text{ This plane is passing through the origin whose position vector is } \vec{a} = \vec{0} \text{ and perpendicular toy-axis whose position vector is } \hat{j} .\]
\[\text{ So, the equation of the XZ-plane is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \hat{j} = \vec{0} . \hat{j} \]
\[ \Rightarrow \vec{r} . \hat{j} = 0\]
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