Changes

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

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.

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).

===See also===

===See also===