Advertisement Remove all ads

Prove that three vectors a,b and c are coplanar, if and only if, there exists a non-zero linear combination xa+yb +zc=0 - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara+ybarb +z barc=0`

Advertisement Remove all ads

Solution

Let `bara, barb, barc ` be coplanar vectors. Then any one of them, say `bara` , will be the linear combination of  `bar b and bar c.`


 there exist scalars α and β such that

`bara=alpha barb +beta barc`

`therefore (-1)bara+alpha barb+beta barc=bar0 `

`i.e " "x bara+y barb+z barc=bar0`

Let x ≠ 0, then divide (1) by x, we get,

`i.e" " bara+(y/x)barb+(z/x)barc=bar0`

`therefore bara=(-y/x)barb+(-z/x)barc`

 i.e. `bara=alpha barb+beta barc, "where " alpha=(-y/x) and beta=-z/x` are scalar

therefore `bara`  is the linear combination of ` bar b and barc.`

Hence, `bara, barb, barc` are coplanar.

Concept: Vector and Cartesian Equations of a Line - Conditions of Coplanarity of Three Vectors
  Is there an error in this question or solution?

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×