Question

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\]\[\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i}  + \hat{j}  + \hat{k}  \right) + \mu\left( 4 \hat{i}  - 2 \hat{j}  + 3 \hat{k} \right)\]

 

Answer 1

` \text{ We know that the equation } \vec{r} = \vec{a} + \lambda \vec{b} + \mu \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} .`
\[\text{ Here} , \vec{a} = \hat{i}  - \hat{j}  + 0 \hat{k}  ; \vec{b} = \hat{i}  + \hat{j}  + \hat{k}  ; \vec{c} = 4 \stackrel\frown is - 2 \hat{j}  + 3 \hat{k}  \]
\[ \text{ Normal vector} , \vec{n} = \vec{b} \times \vec{c} \]
\[ = \begin{vmatrix}\hat{i} & \hat{j}  & \hat{k}  \\ 1 & 1 & 1 \\ 4 & - 2 & 3\end{vmatrix}\]
\[ = 5 \hat{i}  + \hat{j}  - 6 \hat{k}  \]
\[\text{ The vector equation of the plane in scalar product form is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \left( 5 \hat{i} + \hat{j} - 6 \hat{k} \right) = \left( \hat{i} - \hat{j}  + 0 \hat{k}  \right) . \left( 5 \hat{i}  + \hat{j}  - 6 \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( 5 \hat{i}  + \hat{j}  - 6 \hat{k} \right) = 5 - 1 + 0\]
\[ \Rightarrow \vec{r} . \left( 5 \hat{i} + \hat{j} - 6 \hat{k} \right) = 4\]
\[ \Rightarrow \vec{r} . \left( 5 \hat{i} + \hat{j}  - 6 \hat{k}  \right) = 4\]