We know that vectors can be added, subtracted,and
multiplied. Why is the division of vectors is not discussed? Is the
division of vectors possible?
Division of vectors does not yield a definite answer. This is due to the
fact that there are two kinds of multiplication that can produce a vector.
One option is multiplying a vector by a scalar. An example is multiplying a
velocity vector by the number 3. This produces a vector parallel to the
first vector: 3(vectorA)
The other kind of multiplication is the "cross-product", multiply one vector
by another to produce a vector that is perpendicular to both. One
difficulty involves parallel components. Consider:
You can add vector B to vector A and still get the same product:
(vectorA + vectorB)x(vectorB)
The first process produces a product parallel to vectorA. The second
produces a product perpendicular to vectorA. And then there are all the
vectors that are neither parallel nor perpendicular to vector A.
Rather than division, the correct approach is to contemplate what
multiplication could have produced the product, and what restrictions apply.
This is more like saying "What can I multiply by seven to yield 42?", rather
than "What is 42 divided by 7?".
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012