Re: i,j,k things.

[Robert Carlson <Raacc@AOL.COM>]

In a message dated 9/21/99 3:09:59 PM Central Daylight Time,
caviness@SOUTHERN.EDU writes:

> 1.  unit vector = vector with unit length (i.e., length = 1, in the units
>  being used)


I disagree with this statement.  It was always my understanding that a unit
vector was:

1.  A vector.
2.  Had a magnitude of 1.
3.  Is dimensionless.

Below are some definitions of unit vectors from some physics and statics text
books and my comments concerning these definitions.  In general, I find the
physics texts lacking in their definitions and the statics texts more
rigorous.

Serway, Physics for Scientists and Engineers, 3rd Ed.

A unit vector is a dimensionless vector one unit in length used to specify a
given direction.

My comment:  Serway says dimensionless and length in the same breath.  This
is misleading.

Resnick, Halliday, Krane, Physics, 4th Ed.

The unit vectors i, j, and k are used to specify the positive x, y and z axis
respectively.  Each vector is dimensionless and has a length of unity.

My comment:  Same as for Serway's definition.

Feynman, Lectures on Physics

By a unit vector we mean one whose dot product with itself is equal to unity.

My comment:  Not quite sure here.  When Feynman says unity, does he also mean
dimensionless?

Tipler, Physics for Scientists and Engineers, 3rd Ed.

A unit vector is a dimensionless vector that is defined to have the magnitude
of 1 and points in some specified direction.

My comment:  Better.  Dimensionless and no reference to length.

Soutas-Little, Inman, Engineering Mechanics Statics

A unit vector is defined as a vector having a magnitude of unity and a
specific direction.  A unit vector does not have units; that is, its
magnitude is not given in meters, newtons, pounds, etc.

A unit vector may be defined by dividing vector B by a scalar that is equal
to the vector's magnitude.  This yields a unit vector b pointing in the
direction of B.

My comment:  Much better.

Hibbeler, Engineering Mechanics Statics, 6th Ed.

In general, a unit vector is a vector having a magnitude of 1.  If A is a
vector having a magnitude |A| not equal to zero, then a unit vector having
the same direction as A is represented by

A/|A|

Since a vector A is of a certain type, e.g., a force vector, it is customary
to use the proper set of units for its description.  The magnitude of A also
has this same set of units; hence, . . . , the unit vector will be
dimensionless since the units will cancel out.

My comment:  Also much better.

As an example, consider two hooks A and B with a cable connected between
them.  The cable is in tension and we wish to write an expression for the
force that the cable is applying to hook A.  Assume we measure the magnitude
of the tension and get a value of T newtons.  The force then has a magnitude
of T and a direction along the cable from A to B.

If we measure the locations of A and B, then we can find a position vector
from A to B and call it AB.  This position vector has units of length, so, we
can't multiply the tension, T, with AB.  This would give a vector, but would
change the dimensions to those of (force)(length) and would also change the
magnitude.  We wish to multiply the magnitude of the force, T, with a unit
vector that lies in the same direction as the position vector from A to B.

To accomplish this, we find a dimensionless unit vector with a magnitude of 1
that is in the same direction as the position vector from A to B, AB.  It
must be dimensionless so when we multiply it by T, we will still have a
force.  It must also have a magnitude of 1 so when we multiply it by T, the
force vector will still have the magnitude of T.

To find a unit vector in the same direction as AB, we divide AB by the scalar
magnitude of AB, |AB|.  Since vector AB and magnitude |AB| have the same
units, this operation will result in a dimensionless vector in the same
direction as the position vector from A to B and will also have a magnitude
of 1.  We can then write the vector force, T, as:

Vector T = (magnitude of T in units of force)(vector AB in units of
length)/(magnitude of AB in units of length)

This results in a force vector with magnitude T and direction along a line
from A to B.  The unit vector served our purpose, since it was dimensionless
and had a magnitude of 1.

Bob Carlson