If vectors ` vec aa n d vec b` are two adjacent sides of a parallelogram, then the vector respresenting the altitude of the parallelogram which is the perpendicular to `a` is a.` vec b+( vec bxx vec a)/(| vec a|^2)` b. `( vec adot vec b)/(| vec b|^2)` c. ` vec b-( vec bdot vec a)/(| vec a|^2)` d. `( vec axx( vec bxx vec a))/(| vec b|^2)`

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)`