Question

If vec a+vec b+vec c=0. The angle between vec a and vec b and that between vec b and vec c are 150^(@) and 120^(@) respectively. Then the magnitude of vectors.vec a vec b and vec c are in ratio of

Answer 1

The magnitudes of vectors a , b and c are in the ratio of √3 : 2 : 1.

If a + b + c = 0, the angle between a and b and that between b and c are 150° and 120° respectively.

We have to find the magnitude of vectors a , b and c are in ratio.

Here,

⇒ a + b + c = 0

⇒ a + b = - c

Taking modulus both sides,

⇒ |a + b| = |-c|

$\implies\sqrt{a^2+b^2+2abcos150^{\circ}}=|-c|\\\\\implies a^2+b^2-ab\sqrt{3}=c^2\:\:....(1)$

Again,

⇒ a + b + c = 0

⇒ b + c = - a

Taking modulus both sides,

⇒ |b + c| = |-a|

$\implies\sqrt{b^2+c^2+2bccos120^{\circ}}=|-a|\\\\\implies b^2+c^2-bc=a^2\:\:....(2)$

Adding equations (1) and (2) we get,

⇒ 2b² - (ab√3 + bc) = 0

⇒ 2b² = b(a√3 + c)

⇒ b = (a√3 + c)

Here If we put, a = √3k , c = k then b = (√3k × √3 + k) = 2k

Therefore the ratio of a : b : c = √3k : 2k : k = √3 : 2 : 1.

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