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Example:Finding Projection of a Vector

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Explanation:

 

Dot product

 

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 Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.