How may the resultant of two vectors be computed analytically from a vector parallelogram?

The resultant of two vectors is calculated by the formula R^2=P^2+Q^2+2PQcosΘ

The resultant of two vectors can be calculated by the vector parallelogram method, in which basic property of trigonometry is used to setup a simple formula for the resultant of two vectors.

Consider two vector P and Q, making an angle Θ from each other.

From right angled triangle  OCD

OC^2=OD^2+DC^2

R^2=(OA+AD)^2+DC^2

R^2=OA^2+AD^2+2OA*AD+DC^2

From triangle ADC

AC^2=AD^2+CD^2, and cos Θ=AD/AC

R^2=OA^2+AC^2+2OA*AC*cosΘ

R^2=P^2+Q^2+2PQcosΘ