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/
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