How to calculate torque with cross product?

[mohemoto]

Homework Statement

A rod has one end at the origin and one end at the point P whose coordinates are (1m, 2m, 2m). A force F = (3i+2j-1k) N acts on the rod at the point P. What is the torque about the origin due to F?

Homework Equations

torque = F x r

The Attempt at a Solution

I'm not sure what to multiply by what. Are there any suggestions?

 

[rock.freak667]

Well what is the vector OP which is the same as your vector r?

 

[collinsmark]

Use the definition of cross product. (If it's still unclear, do an internet search on "determinant of a matrix.")

$$\vec a \times \vec b = \left| \begin{array}{ccc} \hat \imath & \hat \jmath & \hat k \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{array} \right|$$

 

[PhanthomJay]

Since the force is already given in mutually perpendicular directions x , y, and z, (i, j, and k unit vectors), then use Torque = force times perpendicular distance from line of action of force to the axis about which you are summing moments.
T_x = F_y(z) + F_z(y)
T_y = F_z(x) + F_x(z)
T_z = F_y(x) + F_x(y)

Please watch plus and minus signs. By convention, counterclockwise moments about an axis are taken as positive (x axis points right positive, y-axis points up positive, and z axis points out of plane toward you as positive).

 

[rwisz]

To calculate torque using the cross product, you will need to take the vector cross product of the force vector and the position vector from the origin to point P. This can be written as T = F x r, where T is the torque, F is the force vector, and r is the position vector. In this case, the position vector can be written as r = (1i+2j+2k) m. The cross product of F and r can be calculated using the determinant method or the right-hand rule. Once you have the cross product, the magnitude of the resulting vector will give you the torque about the origin.