Question

given that vector p+q+r=0.|p|=|q|&|r|=√2|p| which of the following can be the angles between vector p,q;vector q,r and vector r,p

Answer 1

angle between P and q is ∅

now,
P + q = -r
take magnitude

p² + q² +2p.qcos∅ = r²
P² +P² +2p.qcos∅= 2P²
cos∅ = 0
∅ =π/2

angle between q and r ∅'
q + r = -P

take magnitude

q² + r² +2rpcos∅'=p²

p² + 2p² +2.√2P.P cos∅' =P²
cos∅' =-1/√2
∅ = 3π/4

angle between r and P is ∅"
P + r = -q

take magnitude

P²+ 2P² +2√2p²cos∅" = P²

cos∅" =-1/√2

∅" = 3π/4

Answer 2

vectors   p + q + r = O
given  |p|  = | q|        and   |r | =  √2 | p |
 Ф = angle between vectors  p and q.

So vector addition  p + q =  - r
           |  p + q |² =  | - r |² = | r |²
           | p |² + | q |²  + 2 | p | *|q| * CosФ =  2 | p |²
           Hence,   cos Ф = 0  =>  Ф = π/2

As p and q vectors of equal magnitude,  - r  will be at  π/4 angle from either p and q.  So r will be at 180 deg from - r.

Angle between  q and r =  180 -45 = 135 deg
 Angle between   vectors r and  p =    135 deg.

We can also find these above angles by  using vector addition formula as given above.