justift whether ( u+v)dot(u cross v)=0 for all non-zero vectors u and v.
need help with this question, managed to finish rest in problem set. thanks for the help
let u = ai + bj + ck
v = di + ej + fk
u + v = (a+d)i + (b+e)j + (c+f)k
u X v = (bf - ce)i - (af - cd)j + (ae - bd)k
(u + v) dot (u X v) =
(a + d)(bf - ce) - (b + e)(af - cd) + (c + f)(ae - bd)
(abf + bdf - ace - cde) - (abf + aef - bcd - cde) + (ace + aef - bcd - bdf)
abf + bdf - ace - cde - abf - aef + bcd + cde + ace + aef - bcd - bdf
believe all these cancel out to 0.
It also helps to know that $\displaystyle <x_1,y_1,z_1>\cdot(<x_2,y_2,z_2>\times<x_3,y_3,z_3 >)$, the "triple product", can be written in the determinant form:
$\displaystyle \left|\begin{array}{ccc}x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3\end{array}\right|$
In the case of either $\displaystyle u\cdot(u\times v)$ or $\displaystyle v\cdot(u\times v)$, two of the rows of the determinant are the same and so the determinant is 0.