151
It’s not hard to imagine a variety of operations that would combine
vectors with vectors or vectors with scalars, but only four of them are
required in order to express Newton’s laws:
operation
definition
vector +vector Add component by component to make
a new set of three numbers.
vector -vector Subtract component by component to
make a new set of three numbers.
vector
.
scalar Multiply each component of the vector
by the scalar.
vector / scalar Divide each component of the vector by
the scalar.
As an example of an operation that is not useful for physics, there just
aren’t any useful physics applications for dividing a vector by another vector
component by component. In optional section 7.5, we discuss in more
detail the fundamental reasons why some vector operations are useful and
others useless.
We can do algebra with vectors, or with a mixture of vectors and scalars
in the same equation. Basically all the normal rules of algebra apply, but if
you’re not sure if a certain step is valid, you should simply translate it into
three component-based equations and see if it works.
Example
Question: If we are adding two force vectors, F+G, is it valid to
assume as in ordinary algebra that F+G is the same as G+F.
Answer: To tell if this algebra rule also applies to vectors, we
simply translate the vector notation into ordinary algebra nota-
tion. In terms of ordinary numbers, the components of the vector
F+G would be F
x
+G
x
, F
y
+G
y
, and F
z
+G
z
, which are certainly the
same three numbers as G
x
+F
x
, G
y
+F
y
, and G
z
+F
z
. Yes, F+G is the
same as G+F.
It is useful to define a symbol r for the vector whose components are x,
y, and z, and a symbol
.
r made out of
.
x,
.
y, and
.
z.
Although this may all seem a little formidable, keep in mind that it
amounts to nothing more than a way of abbreviating equations! Also, to
keep things from getting too confusing the remainder of this chapter
focuses mainly on the
.
r vector, which is relatively easy to visualize.
Self-Check
Translate the equations v
x
=
.
x/
.
t, v
y
=
.
y/
.
t, and v
z
=
.
z/
.
t for motion with
constant velocity into a single equation in vector notation.
v=
.
r/
.
t
Section 7.1Vector Notation