Angle Between Vectors in R2, R3

The dot product also allows us to come up with a convenient formula to find the angle between two vectors. This formula uses the law of cosines, which is a geometric rule valid in R2 and R3 only, but it can be used as a definition to develop further formulas in Rn

In order to find the exact angle in the above example, we can use the following table for some common values for cosθ, in order to find the exact degree or radian value for θ.

By remembering that in a right-angle triangle, cosine = (adjacent/hypotenuse), we can use the following triangles to build the table below.


We can now use this table to find the exact angle between vectors u and v in the previous example,