For any vector `veca` the value of `(vecaxxhati)^2+(vecaxxhatj)^2+(vecaxxhatk)^2` is equal to

Options

A. `vec(a^(2))`
B. `3vec(a^(2))`
C. `4vec(a^(2))`
D. `2vec(a^(2))`

Answer

D

Explanation

Let `veca=xhati+yhatj+zhatk`
`:.vec(a^(2))=x^(2)+y^(2)+z^(2)`
`:.vecaxxhati=|[hati,hatj,hatk],[x,y,z],[1,0,0]|`
`=hati[0]-hatj[-z]+hatk[-y]`
`=zhati-yhatk`
`:.(vecaxxhati)^(2)=(zhatj-yhatk)(zhatj-yhatk)`
Similarly, `(vecaxxhatj)^(2)=x^(2)+z^(2)`
and `(vecaxxhatk)^(2)=x^(2)+y^(2)`
`:.(vecaxxhati)^(2)+(vecaxxhatj)^(2)+(vecaxxhatk)^(2)=y^(2)+z^(2)+x^(2)+z^(2)+x^(2)+y^(2)`
`=2(x^(2)+y^(2)+z^(2))=2veca^(2)`