Vector Arithmetic
In measuring position by distance and direction we used a line
with an arrowhead on it. This representation of position is
called a "vector". Just as there is an arithmetic of
numbers, there is an arithmetic of vectors. We are going to be
interested in motion of particles and have already said that
motion is change in position over time. To get a change in a
number it is customary to subtract the initial value from the
final. To get a change in position we will use the same
technique, subtracting the starting position from the ending. So
how would you subtract two positions? Well it is most convenient
to think of the positions as vectors to do this.
We have identified a line segment with direction and magnitude
as a vector. If V is a particular vector, then
|V| symbolizes its magnitude. Quantities which can be
specified by a magnitude only are called "scalar"
quantities to distinguish them from vectors. Since scalar
quantities are just numbers, they may be added in the ordinary
way. An addition operation is also defined for vectors. Adding a
mixture of vectors and scalars is not defined.
To add vectors there are two techniques available, geometric
addition and algebraic addition. Both yield the same result.
The choice of which technique
to use in adding vectors depends on the application and is a
matter of convenience. First we will discuss geometric vector
addition. Since a vector is defined by its magnitude and
direction, changing its location in our reference frame without
changing its direction or magnitude leaves it the same vector. We
are free to relocate a vector anywhere in our space where we find
it convenient. To add vectors geometrically we just place the
tail of one at the head of the other. The sum then is a vector
from the tail of the first vector to the head of the last. Run
the Geometric Vector Addition display
to see how this works.
The algebraic addition of
vectors involves simply adding up the like components of the
vectors. Imagine a vector with its tail at the origin. The
"scalar components" of that vector are just the
coordinates of the head of the vector. Remember that the
coordinates were the distances along each axis which defined the
position of the head of the vector. To add two vectors, add all
the x components together and all the y components. The algebraic
addition of vectors works because the sums of the components are
the components of the sum. Run the
Algebraic Vector Addition display to see examples of
this.
To break a vector down into its components is called vector
resolution. What that means is that we will resolve a vector into
its "vector components". I have already defined the
scalar components of a vector as the coordinates of the head of
the vector when its tail is at the origin. The vector components
are the vectors lying along the axes which add up to the vector
we are interested in.
Run the Vector
Resolution display to see an example in 3 dimensions. Just
for variety we will use a 3 dimensional vector here. In general
we will use the fewest number of dimensions needed to convey the
desired information but I wanted to illustrate that vectors in 3
dimensions differ from 2 dimensional vectors only in that they
involve an extra coordinate.
Since a vector along an axis has its direction fixed by
definition, only its magnitude changes. That means we can
uniquely define each component of a vector with just a number,
knowing whether we are talking about the x, y or z component.
Remember that we could also specify the position of a particle
with the scalars which we called coordinates. There is an obvious
relationship between the coordinates of a position and the
components of the vector pointing to that position. They are
numerically equal.
Recognize that in vectors as in numbers, subtraction is just
addition with the sign of the entity to be subtracted reversed.
That of course brings up the question of what it means to have
the sign of a vector reversed. To reverse the sign of a number we
just multiply it by -1. To apply the same trick to a vector we
must agree on what it means to multiply a vector by a scalar, in
this case -1.
The product of a scalar times a vector is a vector whose
components are the components of the original vector, each
multiplied by the scalar. Except for some special cases which I
will cover soon, we will use italicized letters to symbolize
vectors. Run the Scalar Vector
Multiplication display to see how this multiplication works.
Verify that when the scalar multiplier is -1, the product vector
has the same magnitude as the original and points the other
way.
Now that we know how to multiply
vectors by scalars we can modify our idea of vector components
slightly. Consider a vector of length 1, in the direction of the
x axis. This is called the x unit vector. There are also similar
y and z unit vectors. Following tradition in these matters we
will label the x unit vector 
(called i hat), the y unit vector
(j hat) and the z unit vector
(k hat). Now if we have
a vector V with its head at coordinates (a,b,c) it can
be expressed as the sum of three vectors a*+b*+c*.
The vector a* is the
product of the unit vector and the scalar component a. The vectors b* and c* are also the products of unit
vectors and their corresponding scalar components. The three
vectors, a*, b* and c* are the vector components of
V. The quantities a, b and c are the scalar components
of V as well as the coordinates of head of the
V vector. The unit vectors are the exception to the
italicizied rule. Run the Vector
Components display to see the ideas of this paragraph
illustrated.
We got into this discussion of multiplying a vector times a
scalar by trying to find a way to subtract vectors. Another way
to look at this subtraction business is to define the negative of
a vector as that vector, which when added to the original yields
a null result. That means the sum is a vector of zero length.
Pretty clearly that is the case if the negative vector has the
same length as the original and points the other way.
One last view of vector
subtraction comes from our discussion of geometric vector
addition. Remember that the sum of two vectors is a vector from
the tail of the first to the head of the second. Now consider a
vector P which is the difference between two vectors,
P2 and P1. This means
P=P2-P1, or by
rearranging the terms,
P2=P1+P. So
P2 must run from the tail of
P1 to the head of P. That makes
P a vector from the head of P1 to the
head of P2. To help visualize why this is so,
Run the Geometric Vector Subtraction
display. We will work in two dimensions since any three vectors
which form a closed figure must be "co-planar" meaning
all in one plane. (Think about it.)
Having added, subtracted, multiplied by a scalar and resolved
vectors, we may as well finish the story with the multiplication
of one vector by another. We will look at two ways to multiply
vectors, both of which have physical significance. First we will
consider the scalar product of two vectors. It is called the
scalar product because the result of the multiplication is a
scalar quantity. This is distinctly different from the scalar
multiplication which we already covered. Sorry about the use of
the word scalar in both cases. In fact that may be why an
alternate name for this kind of multiplication has come into use.
It is also called the "dot product" after the dot
symbol which distinguishes this kind of multiplication. The dot
product of two vectors V1 and
V2 is written V1 ·
V2 where the "·" between the
vectors is the scalar product operator.
To get the dot product of two vectors, multiply the two lengths
together and then multiply by the cosine of the angle between
them. This may be thought of as the product of the length of one
vector times the component of the other vector in the direction
of the first. Run the Dot Product
display to see some examples.
An alternative way to calculate the dot product is to write each vector in terms
of its components and multiply them as you multiply any polynomials in algebra.
For example suppose we had the vectors
The dot product would be:
(4 *
+ 3 *
) (4
*
+ 0 *
) = 4 * 4 *
·
+ 4 * 0 *
·
+ 3 * 4 *
·
+ 3 * 0 *
·
.
Remember that ,
and are vectors perpendicular
to each other, of length 1.0. So the dot product of any different unit vectors
is zero and the dot product of a unit vector with itself is 1. Replacing the unit
vector dot products with their values gives us:
V1 ·
V2 = 4 * 4 * 1 + 4 * 0 * 0 + 3 * 4 * 0 + 3 * 0 *
1 = 16 .
Go back to the Dot Product display to
verify that these two ways of calculating a dot product are
equivalent.
Any terms in our polynomial product
which involve dissimilar unit vectors will be zero because the
unit vectors are perpendicuar and the dot product includes as a
factor the cosine of the angle between the vectors. The dot
product then boils down to just the sum of the products of the
scalar components. In our example above that would be 4 * 4 + 3 *
0. This gives us an easy way to get the dot product of vectors in
three dimensions where the cosine of the angle between the
vectors is not that easy to get a grip on. If
V1 = a *
+ b *
+ c *
, and
V2
= d *
+ e *
+ f *
.
V1 · V2 =
a * d + b * e + c * f .
Now if we want to know the angle between any two vectors in 3D
space, we can divide the dot product by the product of the
lengths and find the angle which has that quotient as its cosine.
We will use |V| to symbolize the length of the vector
V. So (V1 ·
V2)/|V1| *
|V2| is the cosine of the angle between the
vectors. Run the Dot Product 3D
display to see how this works.
I
promised you two ways to multiply vectors together. I know this
seems excessive, two different ways to multiply objects together,
but we are going to need both ways when we come to the ideas of
work and torque.
The second process yields a vector so it is
called the vector product. It is also known as the "cross
product" due to the operation being symbolized by an X, as
in V = V1 X V2
where V is the cross product of the vectors
V1 and V2.
V1 X V2 is defined as a
vector whose length is the product of the lengths of
V1 and V2 times the sine of
the angle between them, and whose direction is perpendicular to
both V1 and V2. The sense
of the cross product is determined by a right hand rule. The
right hand rule in this case works like this. Place the fingers
of your right hand in the direction of V1 and
curl them toward V2. The cross product will
point to the side where your thumb is located. Run the Cross Product display for an illustration.
Notice that V1 X V2 =
-(V2 X V1).
As we did for the dot product, we can calculate the cross product without knowing
explicitly the angle between the vectors. First look at the cross products of
the unit vectors like this:
X
=
X
=
X
= 0,
X
=
,
X
=
,
X
=
To help you remember what equals what in
these relationships, notice that the order of the unit vectors
remains the same in each equation. follows
which follows if you
loop around to the beginning from the end of each equation.
Then suppose
V1 = a *
+ b *
+ c *
and
V2 =
d *
+ e *
+ f *
.
Next multiply V1 X V2
by multiplying each term in V1 by each term
in V2.
V1 X
V2 =
a * d *
X
+ a * e *
X
+ a * f *
X
+
b * d *
X
+ b * e *
X
+ b * f *
X
+
c * d *
X
+ c * e *
X
+ c * f *
X
Next replace all the cross products with their equivalent value.
V1 X
V2 =
a * d * 0 + a * e *
+ a * f * (-
)
+
b * d * (-
) +
b * e * 0 + b * f *
+
c * d *
+ c * e *
(-
) + c * f * 0
Now collect the coefficients of the unit vectors.
V1 X
V2 = (b * f - c * e) *
+ (c * d - a * f) *
+ (a * e - b * d) *
For those of you who remember determinants from your high
school algebra days, you might recognize the formula we just
developed for the cross product as the expansion of a determinant
that looks like this: V1 X V2 =
This is as far as we will go for now with vector arithmetic.
The next section of the course puts much of this vector
arithmetic into action.
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