The moment about point O caused by force F is calculated in vector form simply as 
the cross product of position vector r and force vector F.
Note that r could be any vector as long as it starts from point O and ends anywhere on the line of action of the force. 
Here's a quick review of vector cross product.
Let C be the cross product of two arbitrary vectors A and B. C equals to A cross B.
We can join the tails of the two vectors together 
and then determine the angle between them, theta. 
The magnitude of vector C is determined as the magnitude of vector A times the magnitude of vector B times sine theta. 
That's the magnitude of the cross product. What about the direction of vector C?
The direction is again determined by right hand rule. When you roll your four right hand fingers 
from vector A towards vector B, your thumb points to vector C’s direction.
Vector C is perpendicular to both vector A and vector B,
in other words, vector C is perpendicular to the plane formed by vector A and vector B. 
A cross B is not the same as B cross A. 
In fact, B cross A represents another vector C prime, that is in the opposite direction as vector C,
and therefore equals to negative vector C.
If vectors A and B are given in the Cartesian forms, we can use a matrix to determine the Cartesian form of the cross product of A and B. 
In a 3 by 3 matrix, we fill in i, j, k, and the components of vector A 
and vector B
in this order,
then the cross product equals to 
A_y times B_z, minus A_z times B_y, 
times i,
minus A_x times B_z, minus A_z times B_x, times j,
and lastly plus A_x times B_y, minus A_y times B_x, times k.
Let's look at this example. We know how this force F is directed from point A to point B, we also know its magnitude is 120 pounds,
we need to determine the moment caused by this force about point C in Cartesian vector form.
First we need to determine the force in Cartesian vector form from the position vector r_AB.
And if you recall a similar example in a previous video on position vector and force vector,
I already determined this force vector.
The next step is to find a position vector that starts from the point of interest, in this case point C, and ends on the line of action of 
this force. We have unlimited options, like this one, or this one,
or this one.
And because we have so many choices,
we need to choose a position vector that's the most convenient to work with.
From inspection I choose vector r_CB because it can be easily determined to be 7i foot.
Now we are ready to calculate the moment vector using this vector formulation.
We fill in the matrix: the first row i, j and k, the second row the three components of the 
position vector r_CB, and the last row the three components of the force vector.
Following the calculation formula introduced earlier
we can calculate our answer.
