Question

The vector equation of the plane containing the line \[\vec{r} = \left( - 2 \hat{i} - 3 \hat{j}  + 4 \hat{k}  \right) + \lambda\left( 3 \hat{i}  - 2 \hat{j}  - \hat{k}  \right)\] and the point  \[\hat{i}  + 2 \hat{j}  + 3 \hat{k} \]  is 

 

Options

• \[\vec{r} \cdot \left( \hat{i} + 3 \hat{k}  \right) = 10\]

   

•  \[\vec{r} \cdot \left( \hat{i} - 3 \hat{k} \right) = 10\]

   

•  \[\vec{r} \cdot \left( 3\hat{i} -  \hat{k} \right) = 10\]

• None of these

   

Answer 1

\[\vec{r} \cdot \left( \hat{i} + 3 \hat{k}  \right) = 10\]

\[\text{ Let the direction ratios of the required plane be proportional to   a, b, c .}  \]
\[\text{ Since the required plane contains the line } \vec{r} = \left( - 2 \hat{i} - 3 \hat{j} + 4 \hat{k}  \right) + \lambda \left( 3 \hat{i} - 2 \hat{j} - \hat{k}  \right), \text{ it must pass through the point (-2, -3, 4) and it should be parallel to the line.} \]
\[\text{ So, the equation of the plane is } \]
\[a \left( x + 2 \right) + b \left( y + 3 \right) + c \left( z - 4 \right) = 0 . . . \left( 1 \right) \text{  and }\]
\[3a - 2b - c = 0 . . . \left( 2 \right)\]
\[\text { It is given that plane (1) passes through the point } \hat{i} + 2 \hat{j}  + 3 \hat{k} \text{ or  } (1, 2, 3). \text{ So} ,\]
\[a \left( 1 + 2 \right) + b \left( 2 + 3 \right) + c \left( 3 - 4 \right) = 0 \]
\[3a + 5b - c = 0 . . . \left( 3 \right)\]
\[\text{ Solving (1), (2) and (3), we get } \]
\[\begin{vmatrix}x + 2 & y + 3 & z - 4 \\ 3 & - 2 & - 1 \\ 3 & 5 & - 1\end{vmatrix} = 0\]
\[ \Rightarrow 7 \left( x + 2 \right) + 0 \left( y + 3 \right) + 21 \left( z - 4 \right) = 0\]
\[ \Rightarrow x + 2 + 3z - 12 = 0\]
\[ \Rightarrow x + 3z = 10 \text{ or }  \vec{r} .\left( \hat{i}  + 3 \hat{k} \right)= 10\]