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There are two ways to solve this question :
1) Let the angle between P and q is ∅
now, P + q = -r take magnitude
p² + q² +2p.qcos∅ = r²P² +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
OR
2)Vectors p + q + r = Ogiven |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.
1) Let the angle between P and q is ∅
now, P + q = -r take magnitude
p² + q² +2p.qcos∅ = r²P² +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
OR
2)Vectors p + q + r = Ogiven |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.
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