Question
A force $$\overrightarrow F = 3\hat i + 2\hat j - 4\hat k$$ is applied at the point $$\left( {1,\, - 1,\,2} \right).$$ What is the moment of the force about the point $$\left( {2,\, - 1,\,3} \right)\,?$$
A.
$$\hat i + 4\hat j + 4\hat k$$
B.
$$2\hat i + \hat j + 2\hat k$$
C.
$$2\hat i - 7\hat j - 2\hat k$$
D.
$$2\hat i + 4\hat j - \hat k$$
Answer :
$$2\hat i - 7\hat j - 2\hat k$$
Solution :
Let point $$P$$ is $$\left( {1,\, - 1,\,2} \right)$$
and point $$Q$$ is $$\left( {2,\, - 1,\,3} \right)$$
$$ \Rightarrow $$ Position vector of $$P$$ w.r.t. $$Q$$ is
$$\eqalign{
& \overrightarrow r = \left( {1 - 2} \right)\hat i + \left( { - 1 + 1} \right)\hat j + \left( {2 - 3} \right)\hat k \cr
& \Rightarrow \overrightarrow r = - \hat i + 0\hat j - \hat k{\text{ and }}\overrightarrow F = 3\hat i + 2\hat j - 4\hat k \cr} $$
\[ \Rightarrow {\rm{Moment}} = \overrightarrow r \times \overrightarrow F = \left| \begin{array}{l}
\,\,\,\,\,\hat i\,\,\,\,\,\hat j\,\,\,\,\,\,\,\,\hat k\\
- 1\,\,\,\,0\,\,\, - 1\\
\,\,\,\,\,3\,\,\,\,2\,\,\, - 4
\end{array} \right|\]
$$\eqalign{
& = \hat i\left( {0 + 2} \right) - \hat j\left( {4 + 3} \right) + \hat k\left( { - 2 + 0} \right) \cr
& = 2\hat i - 7\hat j - 2\hat k \cr} $$