The length of the perpendicular from the origin to the plane passing through the point `veca` and containing the line `vecr=vecb+lamdavecc`

Options

A. `([(veca,vecb,vecc)])/(|vecaxxvecb+vecbxxvecc+veccxxveca|)`
B. `([(veca,vecb,vecc)])/(|vecaxxvecb+vecbxxvecc|)`
C. `([(veca,vecb,vecc)])/(|vecbxxvecc+veccxxveca|)`
D. `([(veca,vecb,vecc)])/(|vecaxxvecb+veccxxveca|)`

Answer

C

Explanation

The plane passing through `veca` and containing the line `vecr=vecb+lamdavecc` also passes through the point `vecb` and is parallel to the vector `vecc`. So it is normal to the vector `(veca-vecb)xxvecc`.
Thus, the equation of the plane is
`(vecr-veca).{(veca-vecb)xxvecc}=0`
`implies(vecr-veca).(vecaxxvecc-vecbxxvecc)=0`
`impliesvecr.(vecaxxvecc-vecbxxvecc)=veca.(vecaxxvecc-vecbxxvecc)`
`impliesvecr.(vecaxxvecc-vecbxxvecc)=-veca.(vecbxxvecc)`
`impliesvecr.(vecbxxvecc+veccxxveca)-[(veca,vecb,vecc)]=0`
`:.` Length of the perpendicular fromthe origin to this plane
`=|(vec0.(vecbxxvecc+veccxxveca)-[(veca,vecb,vecc)])/(|vecbxxvecc+veccxxveca|)|=([(veca,vecb,vecc)])/(|vecbxxvecc+veccxxveca|)`