`veca` and `vecb` are two given vectors. On these vectors as adjacent sides a parallelogram is constructed. The vector which is the altitude of the parallelogam and which is perpendicular to `veca` is not equal to

Options

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

Answer

D

Explanation

We have
`AM=` projection of `becb` on `veca=(veca.vecb)/(|veca|)`
`vec(AM)=((veca.vecb)/(|veca|^(2)))veca`
Now in
`vec(AD)=vec(AM)+vec(MD)` or `vec(DM)=vec(AM)-vec(AD)`
`impliesvec(DM)=((veca.vecb)veca)/(|veca|^(2))-vecb`
`(1)/(|veca|^(2)[(veca.vecb)veca-|veca|^(2)vecb]`
`impliesvec(MD)=(1)/(|veca|^(2))[|veca|^(2)vecb-(veca.vecb)veca]`
Now `(vecaxx(vecaxxvecb))/(|veca|^(2))=(1)/(|veca|^(2))[(veca.vecb)veca-(veca.veca)bvecb]`
`vec(DM)`