Dot product

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This article/section deals with mathematical concepts appropriate for late high school or early college.

The dot product is defined for two vectors X and Y to be:

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where |x| is the norm and theta is the angle between the vectors.

Alternatively, and often more conveniently, the dot product of vectors represented by <a,b,c> and <d,e,f> equals a*d + b*e + c*f.

Unlike the cross product, the dot product is a scalar, not a vector, and has no direction.

It follows from the above definition that the dot product of X and Y is 0 if X is perpendicular to Y.

Alternatively, one can describe the dot product as the length of the geometric projection of X onto Y times the length of Y, when the tails of the two vectors are placed at the same point. It must be remembered though, that the dot product is positive if the angle between the two vectors is less than 90 degrees, negative if the angle is between 90 and 180 degrees (it is in this sense that the algebraic sign of the dot product does give some limited directional information).

Application

The dot product is useful in projecting one vector onto another, as in calculating the work done by applying a force to a particle. If you know the dot product of two vectors, then you can easily calculate the angle between the vectors.

The dot product can also be used to find other values through application of the Stokes' Theorem and other theorems.

See also

Cross product