If a vector `veca` is expressed as the sum of two vectors `vec(alpha)` and `vec(beta)` along and perpendicular to a given vector `vecb` then `vec(beta)` is equal to

Options

A. `((vecaxxvecb)xxvecb)/(|vecb|^(2))`
B. `(vecbxx(vecaxxvecb))/(|vecb|^(2))`
C. `(vecbxx(vecaxxvecb))/(|vecb|)`
D. `{(veca.vecb)/((|vecb|)^(2))}vecb`

Answer

B

Explanation

We have `vec(alpha)=` Projection vector of `veca` on `vecbimpliesvec(alpha)=((veca.vecb))/((|vecb|)^(2))vecb`
Clearly `vec(alpha)+vec(beta)=vec(alpha)`
`implies vec(beta)=veca-((veca.vecb))/(|vecb|^(2))vecb`
`impliesvec(beta)=(|vecb|^(2)veca-(veca.vecb)vecb)/(|vecb|^(2))=(vecbxx(vecaxxvecb))/(|vecb|^(2))`