The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
All other sections in this Literature Study Guide are owned and copyrighted by BookRags, Inc.
When placed between two vectors, the dot operator "⋅" signifies that the dot product of the two vectors is to be calculated. This operation is fundamental to matrix multiplication, a computation-intensive task frequently performed in scientific calculations.
The dot product (also known as scalar product or inner product) of two vectors is defined as follows. (Three-dimensional vectors are used for simplicity.) Consider the vectors a = [xa ya za] and b = [xb yb zb]. Their dot product is given by
Note that the dot operator is only defined for two vectors of equal length (or dimensionality), and that its result is not a vector but a scalar (a regular number).
Matrix multiplications can be carried out as a series of dot product operations. For example, multiplying the 3 x 3 matrix M by the 3-dimensional vector d is accomplished as follows. Say that the matrix M consists of three rows, each row a three dimensional vector:
Since each of the three dot products in the column to the right is a scalar number, the column is itself a three-dimensional vector.
It can also be shown that a ⋅ b = |a| |b| cos , where |a| is the length of vector a, |b| is the length of vector |b| and is the angle between them. It is thus simple to determine cos when the vectors a and b are known, and so to determine itself.