(16) |

Most motion is not along a straight line. If fact, almost no motion is
along a line. We therefore need to be able to describe motion along
*multiple dimensions* (usually 2 or 3). That is, we need to be able
to consider and evaluate *vector* trajectories, velocities, and
accelerations. To do this, we must first learn about what vectors are,
how to add, subtract or decompose a given vector in its cartesian
coordinates (or equivalently how to convert between the cartesian,
polar/cylindrical, and spherical coordinate systems), and what scalars
are. We will also learn a couple of products that can be constructed
from vectors.

A **vector** in a coordinate system is a directed line between two
points. It has **magnitude** and **direction**. Once we define a
coordinate origin, each particle in a system has a **position
vector** (e.g. -
) associated with its location in space
drawn from the origin to the physical coordinates of the particle
(e.g. - (
)):

(17) |

The position vectors clearly depend on the choice of coordinate
origin. However, the **difference vector** or **displacement
vector** between two position vectors does **not** depend on the
coordinate origin. To see this, let us consider the **addition** of
two vectors:

(18) |

Note that vector addition proceeds by putting the tail of one at the
head of the other, and constructing the vector that completes the
triangle. To numerically evaluate the sum of two vectors, we
determine their components and add them componentwise, and then
reconstruct the total vector:

(19) | |||

(20) | |||

(21) |

If we are given a vector in terms of its **length** (magnitude)
and **orientation** (direction angle(s)) then we must evaluate its
cartesian components before we can add them (for example, in 2D):

(22) | |||

(23) |

This process is called

The **difference** between two vectors is defined by the addition
law. Subtraction is just adding the negative of the vector in
question, that is, the vector with the **same** magnitude but the
**opposite** direction. This is consistent with the notion of
adding or subtracting its components. Note well: Although the
vectors themselves may depend upon coordinate system, the difference
between two vectors (also called the **displacement** if the two
vectors are, for example, the postion vectors of some particle
evaluated at two different times) does **not**.

When we reconstruct a vector from its components, we are just using
the law of vector addition itself, by **scaling** some special
vectors called **unit vectors** and then adding them. Unit vectors
are (typically perpendicular) vectors that define the essential
directions and orientations of a coordinate system and have unit
length. Scaling them involves multiplying these unit vectors by a
number that represents the magnitude of the vector component. This
scaling number has no direction and is called a **scalar**. Note
that the product of a vector and a scalar is always a vector:

(24) |

where is a scalar (number) and is a vector. In this case, ( is parallel to ).

In addition to multiplying a scalar and a vector together, we can define
products that multiply two vectors together. By ``multiply'' we mean
that if we double the magnitude of either vector, we double the
resulting product - the product is *proportional* to the magnitude
of either vector. There are two such products for the ordinary vectors
we use in this course, and both play *extremely important roles* in
physics.

The first product creates a scalar (ordinary number with magnitude but
no direction) out of two vectors and is therefore called a **scalar
product** or (because of the multiplication symbol chosen) a **dot
product**. A scalar is often thought of as being a ``length''
(magnitude) on a single line. Multiplying two scalars on that line
creates a number that has the *units* of length squared but is
geometrically not an area. By selecting as a direction for that line
the direction of the vector itself, we can use the scalar product to
*define* the length of a vector as the *square root* of the
vector magnitude times itself:

(25) |

From this usage it is clear that a scalar product of two vectors can
never be thought of as an area. If we generalize this idea (preserving
the need for our product to be symmetrically proportional to both
vectors, we obtain the following definition for the general scalar
product:

(26) | |||

(27) |

This definition can be put into *words* - a scalar product is the
length of one vector (either one, say
) times the *component* of the other vector (
) that points
in the *same direction* as the vector
. Alternatively it
is the length
times the component of
parallel to
,
. This product is *symmetric* and *commutative* (
and
can appear in
either order or role).

The other product multiplies two vectors in a way that creates a third
vector. It is called a **vector product** or (because of the
multiplication symbol chosen) a **cross product**. Because a vector
has magnitude and direction, we have to specify the product in such a
way that both are defined, which makes the cross product more
complicated than the dot product.

As far as magnitude is concerned, we already used the non-areal
combination of vectors in the scalar product, so what is left is the
product of two vectors that makes an *area* and not just a ``scalar
length squared''. The area of the parallelogram defined by two vectors
is just:

(28) |

which we can interpret as ``the magnitude of times the component of perpendicular to '' or vice versa. Let us accept this as the magnitude of the cross product (since it clearly has the proportional property required) and look at the direction.

The area is nonzero only if the two vectors do *not* point along the
same line. Since two non-colinear vectors always lie in (or define) a
plane (in which the area of the parallelogram itself lies), and since we
want the resulting product to be independent of the coordinate system
used, one sensible direction available for the product is along the line
*perpendicular to this plane*. This still leaves us with *two*
possible directions, though, as the plane has two sides. We have to
pick one of the two possibilities by *convention* so that we can
communicate with people far away, who might otherwise use a
counterclockwise convention to build screws when we used a clockwise
convention to order them, whereupon they send us left handed screws for
our right handed holes and everybody gets all irritated and everything.

We therefore *define* the direction of the cross product using the
*right hand rule*:

Let the fingers of yourright handlie along the direction of the first vector in a cross product (say below). Let them curl naturally through thesmall angle(observe that there are two, one of which is larger than and one of which is less than ) into the direction of . The erectthumbof your right hand then points in the general direction of the cross product vector - it at least indicates which of the two perpendicular lines should be used as a direction, unless your thumb and fingers are all double jointed or your bones are missing or you used your left-handed right hand or something.

Putting this all together mathematically, one can show that the following are two equivalent ways to write the cross product of two three dimensional vectors. In components:

(29) |

where you should note that appear in

(30) |

or the product

Alternatively in *many* problems it is easier to just use the form:

(31) |

to compute the magnitude and assign the direction

Note that this *axial* property of cross products is realized in
nature by things that *twist* or *rotate around an axis*. A
screw advances into wood when twisted clockwise, and comes out of wood
when twisted counterclockwise. If you let the fingers of your right
hand curl around the screw *in the direction of the twist* your *thumb* points in the direction the screw moves, whether it is in or out
of the wood. Screws are therefore by convention *right handed*.

One final remark before leaving vector products. We noted above that
scalar products and vector products are closely connected to the notions
of *length* and *area*, but mathematics per se need not specify
the *units* of the quantities multiplied in a product (that is the
province of physics, as we shall see). We have numerous examples where
two *different* kinds of vectors (with different units but referred
to a common coordinate system for direction) are multiplied together
with one or the other of these products. In actual fact, there often
*is* a buried squared length or area (which we now agree are
different kinds of numbers) in those products, but it won't always be
obvious in the dimensions of the result.

Two of the most important uses of the scalar and vector product are to
define the *work* done as the force through a distance (using a
scalar product as work is a scalar quantity) and the *torque*
exerted by a force applied at some distance from a center of rotation
(using a vector product as torque is an axial vector). These two
quantities (work and torque) have the *same units* and yet are very
*different* kinds of things. This is just one example of the ways
geometry, algebra, and units all get mixed together in physics.

At first this will be very confusing, but remember, back when you where
in third grade multiplying *integer numbers* was very confusing and
yet rational numbers, irrational numbers, general real numbers, and even
complex numbers were all waiting in the wings. This is more of the
same, but all of the additions will *mean something* and have a
compelling *beauty* that comes out as you study them. Eventually it
all makes very, very good sense.