In mathematics, the dot product is an algebraic operation that
takes two equal-length sequences of numbers (usually coordinate vectors) and returns
a single number obtained by multiplying corresponding entries and then summing
those products. The name is derived from the centered dot "·" that is often used to
designate this operation; the alternative name scalar product emphasizes the scalar (rather than vector) nature of the result.
At a basic level, the dot product is used to obtain the cosine of the angle
between two vectors.
Geometric
interpretation
is the scalar projection of onto .
Since, then .
In Euclidean geometry, the dot
product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector , the dot product is the square of the length
of , or
where denotes the length (magnitude) of . If is another such vector,
where θ is the angle between them.
This formula
can be rearranged to determine the size of the angle between two nonzero
vectors:
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Dot_product
Vector projection
The vector
projection (also known as the vector resolute, or vector component) of a vector in
the direction of a vector (or "of on/onto "), is given by:
where θ is the angle between the vectors and ; the operator is the dot product; and is
the unit vector in the direction of .
(Our solved example in mathguru.com uses
this concept)
The other component of (perpendicular to ), called the vector
rejection of from , is given by:
Both the vector projection and the vector rejection are vectors. The vector projection of on is
the orthogonal projection of onto the straight line defined by . The corresponding vector rejection is the
orthogonal projection of onto a plane orthogonal to .
The vector projection of on can be also regarded as the
corresponding scalar projection multiplied by .
Overview
If A and B are two vectors, the projection of A on B is the vector C with the same direction as B and with the length:
When θ is not known, we can compute using the following property of the dot product :
Thus, the length of C can be also computed as follows
Since C is in the same direction as B,
where is
the unit vector with the same direction as B:
Substituting, we obtain
which is equivalent to either
The latter formula is computationally more efficient than the former,
as the former requires three multiplications and a square root to compute |B| (in turn, needed
to compute ), while the latter only requires three
multiplications to compute (the remaining parts of the two
formulas require the same number and kind of basic algebraic operations).
http://en.wikipedia.org/wiki/Vector_projection
The above explanation is copied from
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