Unit and Basis Vectors in Three Dimensions

Date: 05/09/98 at 15:49:22
From: Louis S. Jagerman
Subject: Definition of unit vector & basis vector

Please give me a simple explanation of:

   1. Unit vector. My books (e.g., _Vector and Tensor Analysis_ by 
      Borisenko) are not clear and assume I already understand this. 
      Also, what use is a unit vector?

   2. Basis vector. Again, my other sources are not clear.

P.S. I study relativity on my own, and this is why I'd like to 
understand the basics, like tensor algebra.

Thanks.

Date: 05/09/98 at 16:39:12
From: Doctor Anthony
Subject: Re: Definition of unit vector & basis vector

A vector is a physical quantity, like velocity, displacement, or 
force, having both magnitude and direction.

Think of a vector as represented by a straight line pointing in a 
particular direction. The length of the line represents the magnitude 
of the vector. So in the case of a unit vector, the length of the line 
is 1 unit.

It is convenient to use unit vectors when working on problems. If we 
let u represent a vector in a certain direction and of unit magnitude, 
then 3u, 7u, and -8u are immediately understandable as vectors of 
magnitudes 3, 7 and -8 all in the direction of u (except -8u, as the 
negative sign means "in the opposite direction to +u").

It is very common to use i, j, and k as unit vectors in the directions 
of the x, y, and z axes, respectively, in 3D space. This means that 
EVERY vector in space can be given in terms of its "components" 
parallel to those three axes. So, for example, 5i + 2j - 6k is a 
vector in space, and its magnitude would be represented by the length 
of a line joining the origin (0,0,0) to the point (5,2,-6). 
Incidentally, this answers your second question: i, j, k are called 
"base" vectors because they are used as the basis for expressing all 
other vectors. Every other vector in 3D space whatsoever can be given 
in terms of i, j, and k.

Sometimes it is convenient to use other vectors as base vectors. Any 
two non-parallel vectors could be used as base vectors to give any 
point in the plane of the two vectors. That is, every other vector in 
that plane could be expressed in terms of the base vectors, just as we 
say 6i + 4j to express a vector in the xy plane. Similarly, any three 
non-coplanar vectors could be used as base vectors "spanning" 3D 
space. Again, the most common base vectors are i,j, k, but there are 
occasions when an entirely different set of base vectors are used.

Finally, vectors are not confined to 1, 2, or 3 dimensions. You can 
have multi-dimensional vectors expressed in terms of 4, 5, 6, and 
higher base vectors. The number of base vectors will equal the 
dimension of the space under consideration.

-Doctor Anthony, The Math Forum
 http://mathforum.org/dr.math/