The three vectors OA,OB,OC have the same magnitude r then the sum of these vectors have magnitude is and the angle between OA and OC is 45 degree and angle between OC and OA is 45 degree

Then angle between vector OB and OA will be 90° now break vector OC into its rectangular components..

Like vector OCcos45° along vector OA and OAsin45° along vector OB then add them simple algebraically where |OA|=|OB|=|OC|=R such as

R+Rcos45° along vector OA

(1+1/√2)R

And

R +Rsin45° along vector OB

(1+1/√2)R

Now apply formula resultant =(A^2 + B^2 + 2ABcos$)^(1/2) where $ is angle between them

Here $ = 90°

So we have sum =(√2 +1)R

Regards

Arun (askIITians forum expert)

ssuming that all three vectors are in the same plane, We can say that angle between OA and OB is 90°. This implies there resultant i.e. OA + OB = root(R^2 + R^2) => root(2) R.

Now the OC vector and resultant vector are in same direction this we can add them algebrically i.e. R + root(2) R

Answer : R [1+root(2)]