Dot productThe product of two vectors is a number. We call it the dot product, or scalar product, of these two vectors. Put the origin of the vectors at the same place.
The dot product of vectors and is the length AB multiplied by the length AH. So it depends on the norm (the length) of the two vectors and on the angle that is between them. If the two vectors form a right angle then the point H is the same than A and the dot product is zero. We say that the 2 vectors are orthogonal if their dot product is zero. If x is the angle between the two vectors, then :
If you know the coordinates of the vectors and if and , hence .
If and , then the two vectors are not orthogonal.
Al Kashi's theoremFirst you have to know that the calculation with the dot product is the same than with a normal product, and you have to notice that so that .
The following relationship is known as Al Kashi's theorem :
If A is a right angle, it gives us Pythagorean theorem.