Q. 95.0( 1 Vote )

Prove that a necessary and sufficient condition for three vectors and to be coplanar is that there exist scalars not all zero simultaneously such that

Answer :

Given: The vectors , and .


To Prove: (a). Necessary condition: The vectors , and will be coplanar if there exist scalar l, m, n not all zero simultaneously such that .


(b). Sufficient condition: For vectors , and , there exist scalar l, m, n not all zero simultaneously such that


Proof:


(a). Necessary condition: Let , and are three coplanar vectors.


Then, one of them can be expressed as a linear combination of the other two.


Then, let


Rearranging them we get,



Here, let


x = l


y = m


–1 = n


We have,



Thus, if , and are coplanars, there exists scalar l, m and n (not all zero simultaneously zero) such that .


necessary condition is proved.


(b). Sufficient condition: Let , and be three vectors such that there exists scalars l, m and n not all simultaneously zero such that .




Now, divide by n on both sides, we get




Here, we can see that


is the linear combination of and .


Clearly, , and are coplanar.


sufficient condition is also proved.


Hence, proved.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Determinants of Matrices of different orderDeterminants of Matrices of different orderDeterminants of Matrices of different order59 mins
Battle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their Graphs48 mins
Triangular Matrices & operations on matricesTriangular Matrices & operations on matricesTriangular Matrices & operations on matrices58 mins
Know About finding the Adjoint & Inverse Of MatrixKnow About finding the Adjoint & Inverse Of MatrixKnow About finding the Adjoint & Inverse Of Matrix46 mins
Complex Numbers | Full chapter QuizComplex Numbers | Full chapter QuizComplex Numbers | Full chapter Quiz55 mins
Determining a determinantDetermining a determinantDetermining a determinant63 mins
Know All About Types of RelationsKnow All About Types of RelationsKnow All About Types of Relations53 mins
Interactive Quiz on Properties of DeterminantsInteractive Quiz on Properties of DeterminantsInteractive Quiz on Properties of Determinants43 mins
Types of Matrices & PropertiesTypes of Matrices & PropertiesTypes of Matrices & Properties51 mins
Lecture on Product of DeterminantsLecture on Product of DeterminantsLecture on Product of Determinants58 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses