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As the angle approaches 0 degrees, the sine of the angle approaches 0, and the cross product of the vectors approaches 0.
As the angle approaches 90 degrees, the sine of the angle approaches 1.0 and the cross product approaches a value that is the product of the magnitudes ofthe two vectors.
Thus, for a given pair of vectors, the cross product can be thought of as a measure of the extent to which they are perpendicular to one another. The closerthey are to perpendicular, the greater will be the value of the cross product.
Hence, the dot product is a measure of the extent to which two vectors are parallel to one another, while the cross product is a measure of the extent towhich two vectors are perpendicular to one another.
Figure 2 contains some facts worth remembering about the vector cross product.
| Figure 2 . Facts worth remembering for the cross product. |
|---|
Given two vectors, A and B with their tails at the origin
AxB = Amag*Bmag*sin(angle between A and B)where
AxB represents the cross product of the vectorsnamed A and B
Amag is the magnitude of vector ABmag is the magnitude of vector B
The cross product is a vector operation. The resultant vectoris perpendicular to the two vectors being multiplied. The
direction of the resultant vector can be determined by theright-hand rule.
The magnitude of the product equals the area of aparallelogram with the vectors forming two sides of the
parallelogram.The cross product is not commutative. In other words, AxB
is not equal to BxA.For a given pair of vectors, the cross product can be
thought of as a measure of the extent to which they areperpendicular to one another. The closer they are to
perpendicular, the greater will be the value of thecross product. |
I encourage you to repeat the computations that I have presented in this lesson to confirm that you get the same results. Then do similarcomputations for pairs of different vectors. For example, swap the positions of vectors A and B and see what this does to the direction of the resultantvector for a cross product.
I will publish a module containing consolidated links to resources on my Connexions web page and will update and add to the list as additional modulesin this collection are published.
This section contains a variety of miscellaneous information.
Financial : Although the openstax CNX site makes it possible for you to download a PDF file for the collection that contains thismodule at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should beaware that some of the HTML elements in this module may not translate well into PDF.
You also need to know that Prof. Baldwin receives no financial compensation from openstax CNX even if you purchase the PDF version of the collection.
In the past, unknown individuals have copied Prof. Baldwin's modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.com showing Prof. Baldwin as the author.Prof. Baldwin neither receives compensation for those sales nor does he know who doesreceive compensation. If you purchase such a book, please be aware that it is a copy of a collection that is freelyavailable on openstax CNX and that it was made and published without the prior knowledge of Prof. Baldwin.
Affiliation : Prof. Baldwin is a professor of Computer Information Technology at Austin Community College in Austin, TX.
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