So let's talk about kinetic energy again.
Remember, when we have an object moving with speed v,
we define the kinetic energy of the object
as a scalar quantity 1/2 mv squared.
This is scalar.
And what's very important to realize,
it is reference frame dependent because the velocity
of the object depends on what reference frame we're in.
One thing to keep in mind is remember our definition
of scalar product, a dot b is the magnitude of a,
the magnitude of v, times the cosine of theta.
So a vector dotted into itself is just
the magnitude of a squared, which
we can just write as a square because the angle is zero.
So here, we have a v squared.
So the dot product b dot b v squared,
enables us to write kinetic energy
as 1/2 m v dot v. And we see our first example of the scalar
product describing a physical quantity.
If we had a coordinate system where
we wrote v as vx i-hat plus vy j-hat plus vz k-hat,
and we took the dot product of that vector with itself,
then we just know that it's the square of the components.
And so, in a Cartesian system, we
can now say that the kinetic energy
is 1/2 m vx squared plus vy squared plus vz squared
because that's the velocity squared.
So there's a simple way to express, abstractly,
the kinetic energy as a dot product
and explicitly in a coordinate system
as the sum of the squares of the components of the velocity.