150
Example (a) shows both ways of writing Newton’s third law. Which
would you rather write.
The idea is that each of the algebra symbols with an arrow written on
top, called a vector, is actually an abbreviation for three different numbers,
the x, y, and z components. The three components are referred to as the
components of the vector, e.g. F
x
is the x component of the vector
F
. The
notation with an arrow on top is good for handwritten equations, but is
unattractive in a printed book, so books use boldface, F, to represent
vectors. After this point, I’ll use boldface for vectors throughout this book.
In general, the vector notation is useful for any quantity that has both
an amount and a direction in space. Even when you are not going to write
any actual vector notation, the concept itself is a useful one. We say that
force and velocity, for example, are vectors. A quantity that has no direction
in space, such as mass or time, is called a scalar. The amount of a vector
quantity is called its magnitude. The notation for the magnitude of a vector
A is |A|, like the absolute value sign used with scalars.
Often, as in example (b), we wish to use the vector notation to repre-
sent adding up all the x components to get a total x component, etc. The
plus sign is used between two vectors to indicate this type of component-
by-component addition. Of course, vectors are really triplets of numbers,
not numbers, so this is not the same as the use of the plus sign with indi-
vidual numbers. But since we don’t want to have to invent new words and
symbols for this operation on vectors, we use the same old plus sign, and
the same old addition-related words like “add,” “sum,” and “total.” Com-
bining vectors this way is called vector addition.
Similarly, the minus sign in example (a) was used to indicate negating
each of the vector’s three components individually. The equals sign is used
to mean that all three components of the vector on the left side of an
equation are the same as the corresponding components on the right.
Example (c) shows how we abuse the division symbol in a similar
manner. When we write the vector
.
v divided by the scalar
.
t, we mean the
new vector formed by dividing each one of the velocity components by
.
t.
(a)F
AonB
=–F
BonA
standsfor
F
AonB,x
=–F
BonA,x
F
AonB,y
=–F
BonA,y
F
AonB,z
=–F
BonA,z
(b)F
total
=F
1
+F
2
+...standsfor
F
total,x
=F
1,x
+F
2,x
+...
F
total,y
=F
1,y
+F
2,y
+...
F
total,z
=F
1,z
+F
2,z
+...
(c)a
=
.
v
.
t
standsfor
a
x
=
.
v
x
/
.
t
a
y
=
.
v
y
/
.
t
a
z
=
.
v
z
/
.
t
Chapter 7Vectors