Question

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\] \[\vec{r} = \left( 1 + s - t \right) \hat{t}  + \left( 2 - s \right) \hat{j}  + \left( 3 - 2s + 2t \right) \hat{k} \]

 

Answer 1

` \text{ The given equation of the plane is } `
\[ \vec{r} = \left( 1 + s - t \right) \hat{i}  + \left( 2 - s \right) \hat{j}  + \left( 3 - 2s + 2t \right) \hat{k}  \]
\[ \Rightarrow \vec{r} = \left( \hat{i}  + 2 \hat{j}  + 3 \hat{k}  \right) + s \left( \hat{i}  - j - 2 \hat{k} \right) + t \left( - \hat{i}  + 0 \hat{j} + 2 \hat{k} \right)\]
\[ \text{ We know that the equation } \vec{r} = \vec{a} + s \vec{b} + t \vec{c} \text{ represents a plane passing through a point whose position vector is } \vec{a} \text{ and parallel to the vectors } \vec{b} \text{ and } \vec{c} .\]
\[\text{ Here } , \vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} ; \vec{b} = \hat{i} - j - 2 \hat{k} ; \vec{c} = - \hat{i} + 0 \hat{j}  + 2 \hat{k} \]
\[\text{ Normal vector } , \vec{n} = \vec{b} \times \vec{c} \]
\[ = \begin{vmatrix}\hat{i}  & \hat{j}  & \hat{k}  \\ 1 & - 1 & - 2 \\ - 1 & 0 & 2\end{vmatrix}\]
\[ = - 2 \hat{i} + 0 \hat{j} - \hat{k} \]
\[ = - 2 \hat{i}  - \hat{k}  \]
\[\text{ The vector equation of the plane in scalar product form is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \left( - 2 \hat{i} - \hat{k}  \right) = \left( \hat{i}  + 2 \hat{j}  + 3 \hat{k}  \right) . \left( - 2 \hat{i}  - \hat{k}  \right)\]
\[ \Rightarrow \vec{r} . \left[ - 1 \left( 2 \hat{i} + \hat{k}  \right) \right] = - 2 + 0 - 3\]
\[ \Rightarrow \vec{r} . \left[ - 1 \left( 2 \hat{i} + \hat{k} \right) \right] = - 5\]
\[ \Rightarrow \vec{r} . \left( 2 \hat{i}  + \hat{k}  \right) = 5\]