Question

$(r.i)(r\times i)+(r.j)(r\times j)+(r.k)(r\times k)$ is equal to

• A. $3r$
• B. $r$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

Let $r=r_{1}i+r_{2}j+r_{3}k$

$\therefore r.i=r_1,r.j=r_2,r.k=r_3$

and $r\times i=0+r_2(j\times i)+r_3(k\times i)=-r_{2}k+r_{3}j$

$\therefore (r.i)(r\times i)=-r_1{r}_{2}k+r_3{r}_{2}j$

Similarly $(r.j)(r\times j)=-r_2{r}_{3}i+r_2{r}_{1}k$

$(r.k)(r\times k)=r_3{r}_{1}j+r_2{r}_{3}i$

$\therefore (r.i)(r\times i)+(r.j)(r\times j)+(r.k)(r\times k)=0$