If i, j, k are orthogonal unit vector triad, then i×(r×i)+j×(r×j)+k×(r×k)=

(i·j) r-(i·r) i+(j·j) r-(j·r) j+(k·k) r-(k·r) k =3r-(i·r) i-(j·r) j-(k·r) kr=xi+yj+zk =3r-xi-yj-zk=3r-r =2r

If i, j, k are orthogonal unit vector triad, then i×(r×i)+j×(r×j)+k×(r×k)=

(i·j) r-(i·r) i+(j·j) r-(j·r) j+(k·k) r-(k·r) k =3r-(i·r) i-(j·r) j-(k·r) kr=xi+yj+zk =3r-xi-yj-zk=3r-r =2r

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If $i, j, k$ are orthogonal unit vector triad, then $i \times (r \times i) + j \times (r \times j) + k \times (r \times k) =$

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$r$

$- r$

$- 2 r$

$2 r$

answer is D.

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Detailed Solution

$\begin{array}{rcl} (i \cdot j) r - (i \cdot r) i + (j \cdot j) r - (j \cdot r) j + (k \cdot k) r - (k \cdot r) k \\ = & 3 r - (i \cdot r) i - (j \cdot r) j - (k \cdot r) k \\ r & = & xi + yj + zk \\ = & 3 r - xi - yj - zk = 3 r - r \\ = & 2 r \end{array}$