Re: unit vector is dimensionless, how to draw when coordinates for length?


Bob Delaney wrote:
In article <1139456528.812781.109060@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<i.love.jeevitha@xxxxxxxxx> wrote:

Say there are x, y, z coordinates set up for "some space" on earth,
where the coordinates represent lengths. Say the space is a
playground or a space around some buildings in downtown new york.

If there is a position vector between 2 points in this space, say
between two buildings or something, then the magnitude of this vector
is a length (metres, or whatever). That is the dimension of the
position vector or any vector which this coordinate system is really
set up for is length.

Now if we find the unit vector of the said position vector, it is
dimensionless. How would one graph the unit vector on this coordinate
system? How would one go about "thinking" about what it really means
to say that this unit vector has magnitude 1? Is that 1m? No. Then what
is it (geometrically) ?

The issues gets even more muddled if we consider forces. Sometimes one
finds the unit vector of a position vector between two points (along a
rope or something) which has a force acting along it. The force vector
can then be determined by multiplying the unit vector by the magnitude
of the force. This obviously means that the unit vector is
dimensionless and can be used to bring about vectors with different
units into the same "x y z" frame. Anyone have an idea about what it
means to say a unit vector has length 1, with respect to thise
coordinate system (which measures lengths)? How can it be graphed in
this xyz frame?


You're correct. Every representation of a vector drawn in the
three-dimensional space we live in, no matter the type of the vector,
is a displacement vector whose components have the dimensions of
length. The vectors can represent force, velocity, acceleration, etc.
Drawn on paper or a blackboard or on a computer screen, they are all
displacement vectors.

Of course if one force vector has twice the magnitude of another force
vector, then the drawn displacement vectors should bear the same
relation. Choose arbitrarily the length of one displacement vector, and
the other displacement vector must be twice that length. And directions
are preserved.


Very good explaination. Explains velocity, force, position vectors
drawn on same axes. But how would you go about explaining dimensionless
vectors (like unit vector) ?

.

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