unit vector of length vector is dimensionless???

snipped-for-privacy@gmail.com Feb 8, 10:42 pm show options
Newsgroups: sci.math From: snipped-for-privacy@gmail.com - Find messages by this author
Date: 8 Feb 2006 19:42:08 -0800 Local: Wed, Feb 8 2006 10:42 pm Subject: unit vector is dimensionless, how to draw when coordinates for length? Reply | Reply to Author | Forward | Print | Individual Message | Show original | Remove | Report Abuse
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?
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Dear i.love.jeevitha:
...

A direction with zero length, if you placed it on a coordinate system with units of length.

How about considering it "up", or "west", or "north"? These have no units of length...

A direction only.
David A. Smith
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i.love.jeevitha wrote:

A unit vector has no units due to how it is calculated (normalized). It simply represents a direction. This isn't rocket science..
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snipped-for-privacy@gmail.com wrote in

It is called the direction cosine of the position vector. It has all sorts of uses in mechanics, none of which I can remember. It is the normalised direction information of that vector and is the complement (in a non mathematical sense) of the magnitude of the vector.
Cheers
Greg
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a vector of length one unit.
How would one go about "thinking" about what it really means

"1" of whatever the coordinate system units are.
Is that 1m?
If the coordinate system is in m, then yes. If it is in feet, then no.
No. Then what

Whatever you choose as your coordinate unit of measure

It is not muddled.
If you have a coordinate system where the unit is newtons, then the unit vector value is one and its units is newtons.

No - it obviously means the unit vector has the units of the coordinate system.
and can be used to bring about vectors with different

see above
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to clarify a bit more 1) - note that you can define scalar and vector elements within a vector, IF you separate and define them. Otherwise, a vector has just direction and magnitude.
2) - you can separate the vector's magnitude from the vector element to get a scalar number value and a vector. These vector components are now standing alone conceptually, without a defined coordinate system. They are still one vector.
to get the unit vector, you divide the vector by the scalar magnitude of the vector.
They are dimensionless, unless I assign them dimensional units.
3) If I place a vector into a coordinate system, both of the components WHICH I HAD CHOSEN TO SEPARATE remain with that vector -, i.e., scalar and unit vector.
The vector placed in a coordinate system assumes the dimensional units of the coordinate system. The unit vector has a scalar value of one. The unit vector placed in a coordinate system has the dimensional units of the coordinate system -either from being a vector in that system or from carrying a scalar number value of one: same-same.
While the following is an incorrect way to look at it, since the vector is not "separated" (even to find the magnitude -any more than you are separated from your height when you describe your height): the unit vector has to have the value of one "in scalar" and one "in vector", or it would not satisfy the identity requirement when one scalar is multiplied by vector element.
4) The unit vector is a vector of value one. Its dimensional value is that of the coordinate system into which it is placed.
In other words, the unit vector sitting in the coordinate system is not actually "splittable" into a scalar and vector quantity, only into conceptual elements which cannot have real number value.
And the unit vector has the dimensions of its coordinate system
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