Home
Class 11
MATHS
If the vectors veca, vecb, and vecc are ...

If the vectors `veca, vecb, and vecc` are coplanar show that `|(veca,vecb,vecc),(veca.veca, veca.vecb,veca.vecc),(vecb.veca,vecb.vecb,vecb.vecc)|=0`

Text Solution

Verified by Experts

Given that `veca, vecb and vecc` are three coplanar vectors. Therefore, there exist scallars x,yand z , we all zone such that
`x veca + yvecb + zvecc = vec0`
Taking dot product of `veca and ` (i), we get
`xveca + yvecb +zvecc=vec0`
Again taking dot product of `vecb and` (i) we get
`xvecb.veca +yvecb.vecb+z vecb.vecc=0`
(i) Now Eqs. (i), (ii) and (iii) form a homogenecous system of equations , where, x , y and z are not all zero, therefore the systems must have a non-trival solution. and for this , the determinant of coefficient matix should be zero , i.e.,
`|{:(veca,vecb,vecc),(veca.veca,veca.vecb,veca.vecc),(vecb.veca,vecb.vecb,vecb.vecc):}|=0`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise 2.1|18 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise 2.2|15 Videos

  • CONIC SECTIONS

    CENGAGE|Exercise Solved Examples And Exercises|1255 Videos

  • LIMITS AND DERIVATIVES

    CENGAGE|Exercise Solved Examples And Exercises|288 Videos

Similar Questions

Explore conceptually related problems

Three vectors veca,vecb and vecc are such that |veca|=2,|vecb|=3,|vecc|=4, and veca+vecb+vecc=vec0. Find 4veca.vecb+3vecb.vecc+3vecc.veca .

Show that : [vecl vecm vecn] [veca vecb vecc]=|(vecl.veca, vecl.vecb, vecl.vecc),(vecm.veca, vecm.vecb, vecm.vecc),(vecn.veca, vecn.vecb, vecn.vecc)|

If veca, vecb, vecc are vectors such that |vecb|=|vecc| then {(veca+vecb)xx(veca+vecc)}xx(vecbxxvecc).(vecb+vecc)=

Three vectors veca,vecb and vecc are such that |veca|=2,|vecb|=3,|vecc|=4 and veca+vecb+vecc=0 find 4veca*vecb+3vecb*vecc+3vecc*veca .

[veca, veca+vecb, veca+vecb+vecc] is :

Prove that [veca-vecb,vecb-vecc,vecc-veca]=0

Three vectors veca, vecb" and "vecc are such that |veca|=2,|vecb|=3,|vecc|=4" and "veca+vecb+vecc=0 find 4veca*vecb+3vecb*vecc+3vecc*veca .

If veca, vecb, vecc are three given non-coplanar vectors and any arbitrary vector vecr in space, where Delta_(1)=|{:(vecr.veca,vecb.veca,vecc.veca),(vecr.vecb,vecb.vecb,vecc.vecb),(vecr.vecc,vecb.vecc,vecc.vecc):}|,Delta_(2)=|{:(veca.veca,vecr.veca,vecc.veca),(veca.vecb,vecr.vecb,vecc.vecb),(veca.vecc,vecr.vec ,vecc.vecc):}| Delta_(3)=|{:(veca.veca,vecb.veca,vecr.veca),(veca.vecb,vecb.vecb,vecr.vecb),(veca.vecc,vecb.vecc,vecr.vecc):}|'Delta=|{:(veca.veca,vecb.veca,vecc.veca),(veca.vecb,vecb.vecb,vecc.vecb),(veca.vecc,vecb.vecc,vecc.vecc):}|, "then prove that " vecr=(Delta_(1))/Deltaveca+(Delta_(2))/Deltavecb+(Delta_(3))/Deltavecc

The vectors veca-vecb,vecb-vecc,vecc-veca are