PROVE THAT SCALAR PRODUCT OF 2 PERPENDICULAR VECTORS IS ZERO

 

 

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 (magnitude) of , denoted by :

If  is another such vector, and  is the angle between them:

This formula can be rearranged to determine the size of the angle between two nonzero vectors:

The Cauchy–Schwarz inequality guarantees that the argument of  is valid.

One can also first convert the vectors to unit vectors by dividing by their magnitude:

then the angle  is given by

As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero. Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length. This property provides a simple method to test the condition of orthogonality.