Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 525

a , respectively. 31

Of course, the time t remains a scalar:

v (t) = v 0 + t a ,
∆ p = v 0 t + (t 2 /2) a ,

Equations for motion
under constant
acceleration, in
vector form

(11.18)

p (t) = p 0 + ∆ p = p 0 + t v 0 + (t 2 /2) a ,

(11.19)

v 0 = ∆ p /t − (1/2)at,
a = 2 ∆ p − t v 0
t 2

.

Note that we didn't make a vector version of Equation (11.17); we'll get to
that in a moment.
This seemingly trivial change in notation is actually hiding two rather
deep facts. First, in the algebraic sense, the vector notation is really just
shorthand for sets of parallel scalar equations for x, y, and z. The important
point is that the three (Cartesian) coordinates are completely independent
of one another. For example, we can make calculations regarding y and
completely ignore the other dimensions, provided that the hypothesis of
constant acceleration is met for the object's motion. If it were not for
the independence of the coordinates, we could not make this change in
notation. The second fact hidden in this notation is that, when we view the
vectors in the equations above as geometric rather than algebraic entities,
the particular coordinate system used to describe those vectors is irrelevant.
We don't even need to specify one. Of course, this is a basic principle
of physics: Mother Nature doesn't know what coordinate system you are
using.
We were able to leap from 1D to 3D mostly just by bolding a few letters
due to the independence of the coordinates. However, there is a bit more
to say about projectile motion in multiple dimensions because there are
situations where we need to consider the effects of all the coordinates at the
same time. One situation has already been alluded to by the lack of a vector
equation corresponding to Equation (11.17). In other words, how could we
solve for time t given a displacement ∆ p , acceleration a , and initial velocity
v 0 ? In one dimension, the projectile is “confined” and basically cannot help
but hitting the target implied by ∆x. 32 But in two or more dimensions, the
situation is more complicated. The increase in complexity that attends the
increase in dimensions is analogous to computing the intersection of two
rays (see Section A.8). In 2D, any two rays must intersect unless they are
parallel, whereas in 3D, the possibility exists for skew rays, which are not
parallel but do not intersect.

31 We use p, which is short for “position,” rather than x, to avoid the assumption that
the x-coordinate is special compared to y or z.
32 With one exception—see Exercise 8.