In the last SparkNote, we discussed position functions in one
dimension. The value of such a function
at a particular time
*t*
_{0}
,
*x*(*t*
_{0})
, was an ordinary number which represented
the position of the object along a single line. In two and three dimensions,
however, the position of an object must be specified by a
vector. We therefore need to upgrade our one-
dimensional function
*x*(*t*)
to
**x**(*t*)
, so that at each moment in time
the position of the object is now given in terms of a vector. Whereas
*x*(*t*)
was a scalar-valued function,
**x**(*t*)
is vector-valued. They
are both, nevertheless, position functions.

As we might expect, the individual components of
**x**(*t*)
correspond to
one-dimensional position functions in each of the two or three directions of
motion. For instance, for motion in three dimensions, the components of
**x**(*t*)
can be labeled
*x*(*t*)
,
*y*(*t*)
, and
*z*(*t*)
, and correspond to
one-dimensional position functions in the
*x*
-,
*y*
-, and
*z*
-directions,
respectively. If we have three-dimensional motion with constant velocity,
**x**(*t*) = **v**
*t*
, where
**v** = (*v*
_{x}, *v*
_{y}, *v*
_{z})
is a
constant vector, the above vector equation for
**x**(*t*)
breaks up into
three one-dimensional equations:

Note that if

What makes the generalization to vectors particularly simple is that the relationships between position, velocity, and acceleration stay exactly the same. Whereas before we had

now we have

where the derivatives are taken

As an example, consider the position function

where

The motion in the

It is important to keep in mind that, although the vector equations for
kinematics look almost identical to their scalar counterparts, the range of
physical phenomena that they can describe is far greater. The last example
suggests that for the same object, completely different motions can be going on
in the
*x*
-,
*y*
-, and
*z*
-directions, even though they are all part of one
overall motion. This idea of breaking up an object's motion into components
will help us analyze two- and three-dimensional motion by using ideas we've
already learned from the one-dimensional case. In the next
section, we put some of these methods to
work when we discuss motion with constant acceleration in more than one
dimension.

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