Triple Scalar Product and Torque Explained?

[Herricane]

Homework Statement

I am working through Boas' Mathematical Methods in the physical sciences book and I don't understand the triple scalar product and torque example.

k [dot] (r X F) = 0 0 1 = xF_y - yF_x
x y z
F_x F_y F_z

k is on the z axis and points in the positive direction. r points in the positive x y and z direction. F points downward in the z direction and it is positive in the x and y direction.

She says "the x and y components of the force can be seen better if we draw them in the (x,y) plane. The torque about the z-axis produced by F_x and F_y is xF_y - yF_x by the elementary definition of torque."

Homework Equations

The Attempt at a Solution

I understand that torque is rFsin theta but I don't understand why it isn't xF_x and yF_y
I don't understand why she is subtracting yF_x from xF_y. They are not negative.

 

[ehild]

I don't understand why she is subtracting yF_x from xF_y. They are not negative.

It is scalar triple product instead of "triple scalar product".
The vector product or cross product is a vector, and its scalar product with a vector is scalar.

The vector product is defined in such way that the product of identical vectors is zero, and it is not commutative, changing the order of the vectors will change the sign of the product. It is easier to understand the way the cross product is calculated if you learn the vector product of the unit vectors along the axes x, y, z: i, j, k.

ixi=jxj=kxk=0,

ixj=k, jxk=i, kxi=j,

and all products with opposite order are of opposite sign.

Two vectors, a and b are

a=a_xi+a_yj+a_zk
and
b=b_xi+b_yj+b_zk,

determine their cross product axb. You need to watch the order of the unit vectors when multiplying them.

I show it in the simpler case, when a_z=0, bz=0.

axb={a_xi+a_yj}x{b_xi+b_yj}=

{(a_xb_x)(ixi)+(a_xb_y)(ixj)+(a_yb_x)(jxi)+(a_yb_y(jxj)=
(a_xb_y-a_yb_x)k,

as ixi=jxj=0 and ixj=k, jxi=-k

ehild