Drive the formula for the magnitude …

If two vectors are considered to be the adjacent sides of a Parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.

The proof for the resultant vector in Parallelogram addition is as follows,

Consider a parallelogram OABC as shown in the figure,

<a href="https://i.stack.imgur.com/gHNkO.jpg" rel="nofollow noreferrer"></a>

Let P&Q be two adjacent sides of parallelogram, and R be the resultant vector obtained by addition of vectors P&Q.

Now, drop a perpendicular from C on OA so that they meet at A

From right angled triangle ΔOCD,

OC²=OD²+DC²

[OD=OA+AD]

R²=(OA+AD)²+DC²

R²=OA²+AD²+2OA.AD+DC²

From ΔADC,

 AC²=AD²+DC²

and also cosθ=AD/AC

And hence R²=OA²+AD²+2OA.AD+DC²

And substituting A and B

R²=A²+B²+2A.Bcosθ