`veca and vecb` are two given vectors. With theses vectors as adjacent sides, a parallelogram is construted. The vector which is the altitude of the parallelogram and which is perpendicular to `veca` is

Options

A. `{((veca.vecb))/(|veca|^(2))}veca-vecb`
B. `1/(|veca|^(2)){(veca.vecb)veca-(veca.veca)vecb}`
C. `(vecaxx(vecaxxvecb))/(|veca|^(2))`
D. `(vecaxx(vecbxxveca))/(|vecb|^(2))`

Answer

D

Explanation

Clearly,

`AM=` Projection of `vecb` on `veca=(veca.vecb)/(|veca|)`
`:. Vec(AM)={(veca.vecb)/(|veca|^(2))}veca`
In `DeltaADM`, we have
`vec(AD)=vec(AM)+vec(MD)`
`impliesvec(DM)=vec(AM)-vec(AD)impliesvec(DM)=((veca.vecb)veca)/(|veca|^(2))-vecb`
Also
`vec(DM)=1/(|veca|^(2)){(veca.vecb)veca-|veca|^(2)vecb}=1/(|veca|^(2)){(veca.vecb)veca-(veca.veca)vecb}`
`=vec(DM)=(vecaxx(vecaxxvecb))/(|veca|^(2))`
Hence vector in option d is equal to the required vector.