ACTUALLY, you
CAN multiply two vectors, either by dot product (which gives a scalar product) or the cross product (which gives you a perpendicular vector product). SO, now that we established that two vectors can be multiplied, we can conclude that in the equation,
vf^2 =
vi^2 + 2
ad,
acceleration is a vector and
"d" refers to displacement. The question that remains is, what kind of vector multiplication is involved here? Obviously,
vf =
vf so they're heading in the same linear direction, indicating that a cross product could give two possible solutions (since we work in three dimensions). Now, such ambiguity cannot be tolerated in physics! As well, squaring a value yields the same solution, regardless of sign (ie. positive times positive equals positive, negative times negative equals positive), so it becomes apparent that multiplying two vectors in this physical equation involves a
dot product to yield a scalar quantity.
So how does this proof relate to my original question of the nature of time? First, we must refer back to one of Cindy's equations,
vf =
vi +
at. We know that
vf,
vi, and
a are vectors, meaning that
vi +
at =
vector. And we know that the vector multiplication we are concerned with in physics is a dot product. If t was indeed a vector, then
vi +
at =
vector + scalar, which does not equate vector, indicating t would have to be a scalar for the equation to hold true. Otherwise, the world of elementary mechanics would spin on its head!!!!
To consolidate the fact that time works in one direction, it may help to refer to gravity. Yes, gravity works in one direction (it's towards the centire of mass of a body, ie. not just down!), but it's a vector since it's a force. Recall that
F = m
a. Note that as we established before, acceleration is a vector, and it's not disputed that mass is a scalar quantity. Therefore, it's very likely that the concept of forward-moving time is a completely different function altogether (hereafter referred to as
X) that involves multiplying the scalar quantity time with another vector that indicates directionality (hereafter referred to as
b). Thus, the progression of time can be expressed as
X = t
b.
So perhaps that's why we have not yet succeeded in building a "time" machine, since there's no direction in time to travel in. Rather, we should turn our attentions to an "unknown function machine."