If vectors `veca and vecb` are two adjacent sides of parallelograsm then the vector representing the altitude of the parallelogram which is perpendicular to `veca` is (A) `vecb+(vecbxxveca)/(|veca|^2)` (B) `(veca.vecb)/(vecb|^2)` (C) `vecb-(vecb.veca)/(|veca|)^2)` (D) `(vecaxx(vecbxxveca))/(vecb|^20`

Options

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

Answer

a,b,c

Explanation

w have
AM = projection of `vecb on veca = (veca . Vecb)/(|vecb|)`
`vec(AM) = ((veca.vecb)/(|veca|^(2)))veca` ,
Now, in `triangle ADM`
`vec(AD) =vec(AM) +vec(MD) or vec(DM)=vec(AM)-vec(AD)`
`Rightarrow vec(DM)= ((veca.vecb)veca)/(|veca|^(2))-vecb`
` = 1/(|veca|^(2))[(veca.vecb)veca-|veca|^(2) vecb]`
`vec(MD) = 1/(|veca|^(2))[|veca|^(2)vecb- (veca.vecb)veca]`
Now, ` (vecaxx (vecaxxvecb))/(|veca|^(2))=1/(|veca|)[(veca.vecb)veca-(veca.vecb)vecb]`
`vec(DM)`