Question

Find the vector equation of the plane \[\overline{r}=\left( 2\widehat{i}+\widehat{k} \right)+\lambda \widehat{i}+\mu \left( \widehat{i}+2\widehat{j}-3\widehat{k} \right)\] in a scalar product form.

Answer 1

Hint: In this type of question we have to use the concept of vectors. We know that the equation \[\overline{r}=\overline{a}+\lambda \overline{b}+\mu \overline{c}\] represents a plane passing through a point whose position vector is \[\overline{a}\]. Also we know that the vector equation of the plane in scalar product form is \[\overline{r}\cdot \overline{n}\] where \[\overline{n}\] is the normal vector which is given by, \[\overline{n}=\overline{b}\times \overline{c}\]. We can derive the formula of cross product of two vectors \[\overline{A}=a\widehat{i}+b\widehat{j}+c\widehat{k}\] and \[\overline{B}=x\widehat{i}+y\widehat{j}+z\widehat{k}\] as \[\overline{A}\times \overline{B}=\left| \begin{matrix}
   \widehat{i} & \widehat{j} & \widehat{k} \\
   a & b & c \\
   x & y & z \\
\end{matrix} \right|\]. Also we can derive the formula of dot product of two vectors \[\overline{A}=a\widehat{i}+b\widehat{j}+c\widehat{k}\] and \[\overline{B}=x\widehat{i}+y\widehat{j}+z\widehat{k}\] as \[\overline{A}\cdot \overline{B}=\left( a\times x \right)+\left( b\times y \right)+\left( c\times z \right)\].

Complete step by step answer:
Now we have to find the vector equation of the plane \[\overline{r}=\left( 2\widehat{i}+\widehat{k} \right)+\lambda \widehat{i}+\mu \left( \widehat{i}+2\widehat{j}-3\widehat{k} \right)\] in a scalar product form.
We know that the equation \[\overline{r}=\overline{a}+\lambda \overline{b}+\mu \overline{c}\] represents a plane passing through a point whose position vector is \[\overline{a}\].
Here, \[\overline{r}=\left( 2\widehat{i}+\widehat{k} \right)+\lambda \widehat{i}+\mu \left( \widehat{i}+2\widehat{j}-3\widehat{k} \right)\] comparing it with \[\overline{r}=\overline{a}+\lambda \overline{b}+\mu \overline{c}\] we get,
\[\begin{align}
  & \Rightarrow \overline{a}=\left( 2\widehat{i}+\widehat{k} \right) \\
 & \Rightarrow \overline{b}=\widehat{i} \\
 & \Rightarrow \overline{c}=\left( \widehat{i}+2\widehat{j}-3\widehat{k} \right) \\
\end{align}\]
Also we know that the vector equation of the plane in scalar product form is \[\overline{r}\cdot \overline{n}\] where \[\overline{n}\] is the normal vector which is given by, \[\overline{n}=\overline{b}\times \overline{c}\].
Hence, the normal vector, \[\overline{n}=\overline{b}\times \overline{c}\] is given by,
\[\begin{align}
  & \Rightarrow \overline{n}=\overline{b}\times \overline{c} \\
 & \Rightarrow \overline{n}=\left| \begin{matrix}
   \widehat{i} & \widehat{j} & \widehat{k} \\
   1 & 0 & 0 \\
   1 & 2 & -3 \\
\end{matrix} \right| \\
 & \Rightarrow \overline{n}=\widehat{i}\left( 0-0 \right)-\widehat{j}\left( -3-0 \right)+\widehat{k}\left( 2-0 \right) \\
 & \Rightarrow \overline{n}=3\widehat{j}+2\widehat{k} \\
\end{align}\]
Also as we know that, the vector equation of the plane in scalar product form is given by \[\overline{r}\cdot \overline{n}=\overline{a}\cdot \overline{n}\]
Hence, we can write
\[\begin{align}
  & \Rightarrow \overline{r}\cdot \overline{n}=\overline{a}\cdot \overline{n} \\
 & \Rightarrow \overline{r}\cdot \left( 3\widehat{j}+2\widehat{k} \right)=\left( 2\widehat{i}+\widehat{k} \right)\cdot \left( 3\widehat{j}+2\widehat{k} \right) \\
 & \Rightarrow \overline{r}\cdot \left( 3\widehat{j}+2\widehat{k} \right)=\left( 2\widehat{i}+0\widehat{j}+\widehat{k} \right)\cdot \left( 0\widehat{i}+3\widehat{j}+2\widehat{k} \right) \\
 & \Rightarrow \overline{r}\cdot \left( 3\widehat{j}+2\widehat{k} \right)=\left( 2\times 0 \right)+\left( 0\times 3 \right)+\left( 1\times 2 \right) \\
 & \Rightarrow \overline{r}\cdot \left( 3\widehat{j}+2\widehat{k} \right)=2 \\
 & \Rightarrow \overline{r}\cdot \left( 3\widehat{j}+2\widehat{k} \right)-2=0 \\
\end{align}\]
Hence, \[\overline{r}\cdot \left( 3\widehat{j}+2\widehat{k} \right)-2=0\] is the required vector equation of the plane \[\overline{r}=\left( 2\widehat{i}+\widehat{k} \right)+\lambda \widehat{i}+\mu \left( \widehat{i}+2\widehat{j}-3\widehat{k} \right)\] in scalar product form.

Note: In this type of question students have to remember the vector equation of the plane \[\overline{r}\] in scalar form which is given by \[\overline{r}\cdot \overline{n}=\overline{a}\cdot \overline{n}\]. Also students have to remember the formulas of cross product and dot product of two vectors. Students have to note that in evaluation of cross product of two vectors they have to use the concept of evaluation of determinants while in case of dot product of two vectors they have to perform sum of the multiplication of the coefficients of the unit vectors i.e. \[\widehat{i}\], \[\widehat{j}\] and \[\widehat{k}\].