Question

Line \[\mathop r\limits^ \to {\text{ }} = \left( {{\text{ }}i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right){\text{ }} + {\text{ }}t{\text{ }}\left( {{\text{ }}2i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right)\] contained in a plane to which vector \[\mathop n\limits^ \to {\text{ }} = {\text{ }}3i{\text{ }} - {\text{ }}2j{\text{ }} + {\text{ }}\lambda k\] is normal . Find the value of \[\lambda \] . Also find the vector equation of the plane .

Answer 1

Hint: We have to find the value of \[\lambda \] and the vector equation of the plane . We solve this question using the concept of three dimensional geometry and vector algebra and we also use the concept of two vectors perpendicular to each other . We use the formula of two vectors perpendicular to each other to find the value of \[\lambda \] and then find the vector equation of the plane using the formula for the vector equations .

Complete step-by-step answer:
Given :
 \[\mathop r\limits^ \to {\text{ }} = {\text{ }}\left( {{\text{ }}i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right){\text{ }} + {\text{ }}t{\text{ }}\left( {{\text{ }}2i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right)\]
 \[\mathop r\limits^ \to {\text{ }} = {\text{ }}\left( {{\text{ }}2t{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }}i{\text{ }} - {\text{ }}\left( {{\text{ }}t{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }}j{\text{ }} + {\text{ }}\left( {{\text{ }}t{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }}k\]
 \[\mathop n\limits^ \to {\text{ }} = {\text{ }}3i{\text{ }} - {\text{ }}2j{\text{ }} + {\text{ }}\lambda k\]
As given , the line \[\mathop r\limits^ \to {\text{ }} = {\text{ }}\left( {{\text{ }}i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right){\text{ }} + {\text{ }}t{\text{ }}\left( {{\text{ }}2i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right)\] is normal to \[\mathop n\limits^ \to {\text{ }} = {\text{ }}3i{\text{ }} - {\text{ }}2j{\text{ }} + {\text{ }}\lambda k{\text{ }}.\]
This means that \[\mathop r\limits^ \to {\text{ }}\] and \[\mathop n\limits^ \to {\text{ }}\] are perpendicular to each other .
We also know that the dot product to two vectors is zero if both the vectors are perpendicular .
So , we get
 \[\left( {{\text{ }}2i{\text{ }} - {\text{ }}j{\text{ }} + {\text{ }}k{\text{ }}} \right){\text{ }}.{\text{ }}\left( {{\text{ }}3i{\text{ }} - {\text{ }}2j{\text{ }} + {\text{ }}\lambda k{\text{ }}} \right){\text{ }} = {\text{ }}0\]
As , we know that
 \[i{\text{ }}.{\text{ }}i{\text{ }} = {\text{ }}j{\text{ }}.{\text{ }}j{\text{ }} = {\text{ }}k{\text{ }}.{\text{ }}k{\text{ }} = {\text{ }}1\] Also ,
i . j = i . k = j . k = 0
By multiplication of terms , we get
 \[6{\text{ }} + {\text{ }}2{\text{ }} + {\text{ }}\lambda {\text{ }} = {\text{ }}0\]
We get ,
 \[\lambda {\text{ }} = {\text{ }} - 8\]
The points that lies on the plane are \[\left( {{\text{ }}1{\text{ }},{\text{ }} - 1{\text{ }},{\text{ }}1{\text{ }}} \right)\]
Using this points , we get values of points as :
 \[\left( {{\text{ }}x{\text{ }} - {\text{ }}1{\text{ }}} \right){\text{ }},{\text{ }}\left( {{\text{ }}y{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }},{\text{ }}\left( {{\text{ }}z{\text{ }} - {\text{ }}1{\text{ }}} \right)\]
Vector equation of plane is given as :
Putting these points in \[\overrightarrow n \] , we get
 \[3{\text{ }}\left( {{\text{ }}x{\text{ }} - {\text{ }}1{\text{ }}} \right){\text{ }} - {\text{ }}2{\text{ }}\left( {{\text{ }}y{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }} + {\text{ }}\lambda {\text{ }}\left( {{\text{ }}z{\text{ }} - {\text{ }}1{\text{ }}} \right){\text{ }} = {\text{ }}0\]
putting the value of $\lambda $ , we get
 \[3{\text{ }}\left( {{\text{ }}x{\text{ }} - {\text{ }}1{\text{ }}} \right){\text{ }} - {\text{ }}2{\text{ }}\left( {{\text{ }}y{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }} - {\text{ }}8{\text{ }}\left( {{\text{ }}z{\text{ }} - {\text{ }}1{\text{ }}} \right){\text{ }} = {\text{ }}0\]
Expanding the terms , we get
 \[3{\text{ }}x{\text{ }} - {\text{ }}2{\text{ }}y{\text{ }} - {\text{ }}8{\text{ }}z{\text{ }} + {\text{ }}3{\text{ }} = {\text{ }}0\]
Thus , the value of \[\lambda \] is \[ - 8\] and the vector equation of the plane is \[3{\text{ }}x{\text{ }} - {\text{ }}2{\text{ }}y{\text{ }} - {\text{ }}8{\text{ }}z{\text{ }} + {\text{ }}3{\text{ }} = {\text{ }}0{\text{ }}.\]

Note: The scalar product of two given vectors \[\mathop a\limits^ \to {\text{ }}\] and \[\mathop b\limits^ \to {\text{ }}\] having angle $\theta $ between them is defined as :
 \[\mathop a\limits^ \to {\text{ }}.{\text{ }}\mathop b\limits^ \to {\text{ }} = {\text{ }}\left| a \right|{\text{ }}\left| b \right|{\text{ }}cos{\text{ }}\theta \] Also , when \[\mathop a\limits^ \to {\text{ }}.{\text{ }}\mathop b\limits^ \to {\text{ }}\] is given , the angle ‘$\theta $’ between the vectors \[\mathop a\limits^ \to {\text{ }}\] and \[\mathop b\limits^ \to {\text{ }}\].