Text: One can also define an operation on vectors (of all dimensions) which we shall, for the moment, call a change of scale [magnitude]. Consider the displacement vector (3,4,12). The same may represent the vector (6,8,24), whose components are twice as great, or the vector (1,4/3,4), whose components are one-third as great. It would be merely a matter of changing the scale indicated. The change of scale is tantamount to multiplying a vector by an ordinary number, and is accomplished by multiplying each component of the vector by this number. This rule for multiplying a vector by a number has been illustrated for positive rational numbers only. It has more general validity, however, but the description as a change of scale may not seem appropriate if one multiplies the vector by zero, in which case it is converted into a null or zero vector. Again, to multiply the vector by a negative number would require a change of scale that reverses the signs of numbers on the X-, Y-, and Z-axes. In such cases, an alternative interpretation of the multiplication is possible, where instead of changing the scale one changes the vector. Then multiplying it by a positive number would stretch or contract it, according to whether this number is greater than or less then one, and multiplication by a negative would not only stretch or contract it, but would also reverse the direction of the vector. The reader may find the description of multiplication of a vector by a number confusing if the vector is also considered a sort of "number" albeit a complex or hypercomplex number. Therefore physicists apply the term scalar to the single "ordinary number," like 2 or 1/3, which effects the change of scale. Then what we have been talking about is the multiplication of a vector by a scalar, that is, multiplication of a complex or hypercomplex number by a single "ordinary number." As far as vectors are concerned, scalar multiplication is a unary operation because it is carried out on one vector only. It is a change of scale in the components of that one vector or, in the alternative interpretation, a stretching or shrinking of that vector.

See Also: QUANTUM ARITHMETIC; PLANE

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