So, if
A = A i + A j + A k
x
y
z
and
B = B i + B j + B k
x
y
z
then A ⋅ B is just
A ⋅ B = (A i + A j + A k) ⋅(B i + B j + B k)
x
y
z
x
y
z
A ⋅ B = A i ⋅(B i + B j + B k) +
x
x
y
z
A j ⋅ (B i + B j + B k) +
y
x
y
z
A k ⋅ (B i + B j + B k)
z
x
y
z
A ⋅ B = A i ⋅ B i + A i ⋅ B j + A i ⋅ B k +
x
x
x
y
x
z
A j ⋅ B i + A j ⋅ B j + A j ⋅ B k +
y
x
y
y
y
z
A k ⋅ B i + A k ⋅ B j + A k ⋅ B k
z
x
z
y
z
z
A ⋅ B = A B i ⋅i + A B i ⋅j + A B i ⋅k +
x
x
x
y
x
z
A B j ⋅i + A B j ⋅j + A B j ⋅k +
y
x
y
y
y
z
A B k ⋅i + A B k ⋅j + A B k ⋅k
z
x
z
y
z
z
A ⋅ B = A B + A B + A B
x
x
y
y
z
z
The end result is that the dot product of two vectors is simply the sum of: the product of the two
vectors’ x components, the product of their y components, and the product of their z components.
Energy Transfer Work vs. Center of Mass Pseudo-Work
I introduced the topic of work by stating that it represents one category of energy transfer to a
system. As such, work is energy transfer work. There is a quantity that is calculated in much the
same way as work, with one subtle difference. I’m going to call the quantity center of mass
pseudo-work and I’m going to use a couple of specific processes involving a frictionless
horizontal surface, a spring, and a block (and in one case, another block) to distinguish energy
transfer work from center of mass pseudo-work. Suppose we attach the spring to the wall so that
the spring sticks out horizontally and then push the block toward the wall in such a manner as to
compress the spring. Then we release the block from rest and start our observations at the first
instant subsequent to release. Let our system be the block. The spring pushes the block away
from the wall. The spring transfers energy to the block while the spring is in contact with the
block. The work done can be calculated as the integral of vector force dot vector infinitesimal
displacement which I'll loosely state as the integral of force times distance. The distance in this
case is the displacement (the infinite set of infinitesimal displacements) of the point of
application of the force. This kind of work is energy transfer work. It is the amount of energy
transferred to the block by the spring.
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