It is defined as:
where θ is the angle between the two vectors. Thus, the dot product of two perpendicular vectors is always zero. If a and b are both unit vectors (ie of length 1), the the dot product simply gives the cosine of the angle between them. Thus, given two vectors, the angle between them can be found by rearranging the above formula:
\left({ \mathbf{a \cdot b} \over \mathbf{a b} }\right)</math>
The dot product is particularly used in resolution of forces[?]. If b is a unit vector, then the dot product gives the projection of a in direction b. In mechanics, this gives the component of a force in that direction.
Work is the dot product of force and displacement.
Properties The definition has the following consequences:
From these it follows directly that the dot product of two vectors a = [a_{1} a_{2} a_{3}] and b = [b_{1} b_{2} b_{3}] given in coordinates can be computed particularly easily:
or, using matrix multiplication and treating the vectors as 1by3 matrices:
The dot product satisfies all the axioms of an inner product. In an abstract vector space, the notion of angle between the elements of the space can be defined in terms of the inner product.
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