Question

If a+b+c=0 and |a|=3,|b|=5 and |c|=7 then angle between a and b

Answer 1

it is given that, a + b + c = 0

⇒a + b = -c

⇒|a + b | = |-c| = |c|

magnitude of |a + b| is given as √{|a|² + |b|² + 2|a||b|cosθ} , where θ is angle between a and b.

now, √{|a|² + |b|² + 2|a||b|cosθ} = |c|

squaring both sides,

⇒|a|² + |b|² + 2|a||b|cosθ = |c|²

putting values of |a|, |b| and |c|.

⇒(3)² + (5)² + 2(3)(5)cosθ = 7²

⇒9 + 25 + 30cosθ = 49

⇒34 + 30cosθ = 49

⇒30cosθ = 49 - 34 = 15

⇒cosθ = 15/30 = 1/2

⇒cosθ = cos30°

⇒θ = 30°

hence, angle between a and b is 30°