Question

A force of $ - F\hat k$ acts on O, the origin of the coordinate system. The torque about the point (1,-1) is
A) $F\left( {\hat i - \hat j} \right)$
B) $ - F\left( {\hat i + \hat j} \right)$
C) $F\left( {\hat i + \hat j} \right)$
D) $ - F\left( {\hat i - \hat j} \right)$

Answer 1

Hint: We can use the formula of torque which is the cross product of lever arm and force defined as torque or moment of force.
Using this formula we can get the desired torque.

Complete step-by-step answer:
Torque is defined as it is a turning effect of force about the axis of rotation. It is measured as the magnitude of force and the perpendicular distance between the line of action of the force and the axis of rotation.
Torque = force ×perpendicular distance between force and axis of rotation
Or it can be defined as. The torque acting on the particle is defined as the vector product of position and force vector.
$ \Rightarrow \vec \tau = \vec r \times \vec F$
In this question given $\vec F = - F\hat k$
Torque about point (1,-1) convert it into vector form $\vec r = 1\hat i - 1\hat j$
Now use above formula
$ \Rightarrow \vec \tau = (1\hat i - 1\hat j) \times - F\hat i$
Solving this
$\vec \tau = \left( {1\hat i \times - F\hat k} \right) + \left( { - 1\hat j \times - F\hat k} \right)$
As we know $\hat i \times \hat k = - \hat j$ and $\hat j \times \hat k = \hat i$
$\vec \tau = F\hat j + F\hat i$
Simplify it
$\therefore \vec \tau = F\left( {\hat j + \hat i} \right)$

Hence option C is correct

Note: As you can see above we use cross product of $\vec r$ and $\vec F$ in cross product the cross product of unit vector $\hat i,\vec j,\hat k$ can be defined as
 \[
  \hat i \times \hat j = \hat k \\
  \hat j \times \hat k = \hat i \\
  \hat k \times \hat i = \hat j \\
\]
If we reverse cross product then minus sign comes in answer as
\[
  \hat j \times \hat i = - \hat k \\
  \hat k \times \hat j = - \hat i \\
  \hat i \times \hat k = - \hat j \\
\]