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Example:Finding Angle Between Two Vectors

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Euclidean vector


Illustration of a vector


A vector going from A to B

In elementary mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or - as here - simply a vector) is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by 




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)




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.