Question
A force $$F = 2i + j - k$$ acts at a point $$A,$$ whose position vector is $$2i - j.$$ The moment of $$F$$ about the origin is :
A.
$$i + 2j - 4k$$
B.
$$i - 2j - 4k$$
C.
$$i + 2j + 4k$$
D.
$$i - 2j + 4k$$
Answer :
$$i + 2j + 4k$$
Solution :
Force $$\left( {\vec F} \right) = 2i + j - k$$ and its position vector of $$A = 2i - j.$$ We know that the position vector of a force about origin $$\left( r \right) = \left( {2i - j} \right) - \left( {0i + 0j + 0k} \right)$$ or $$r = 2i - j.$$
Therefore, moment of the force about origin
\[ = r \times \overrightarrow F = \left| \begin{array}{l}
i\,\,\,\,\,\,\,\,\,j\,\,\,\,\,\,\,\,\,\,\,k\\
2\,\,\, - 1\,\,\,\,\,\,\,\,0\\
2\,\,\,\,\,\,\,\,1\,\,\,\, - 1
\end{array} \right| = i + 2j + 4k.\]