Question

If a, b, c are three vectors such that a+b+c = 0. The angle between a and b is 150° and the angle between b and c is 120° . Find the ratio of magnitude of a, b and c.

Answer 1

Answer:Here's your answer

Explanation:

a + b + c = 0

or, a + b = - c

or, |a + b | = |-c |

or, √(a² + b² + 2abcosα) = c²

but given, angle between a and b = 150°

so, α = 150°

or, √{a² + b² + 2abcos150°} = c²

squaring both sides,

or, a² + b² + 2abcos150° = c²

or, a² + b² - √3ab = c² ........(1)

again, a + b + c = 0

or, b + c = -a

or, |b + c| = |-a|

or, √(b² + c² + 2bccosβ) = a²

but angle between b and c is 120° .

so, β = 120°

or, √(b² + c² + 2bccos120°} = a²

squaring both sides,

or, b² + c² + 2bccos120° = a²

or, b² + c² - ab = a² .......(2)

from equations (1) and (2),

2b² - (√3 + 1)ab = 0

or, 2b² = (√3 + 1)ab

or, b = {(√3 + 1)/2}a .......(3)

from equation (1),

a² + (√3 + 1)²/4 a² - √3a²(√3 + 1)/2 = c²

or, a² [ 1 + (4 + 2√3)/4 - (√3 + 3)/2] = c²

or, a² {4 + 4 + 2√3 - 2√3 - 6}/4 = c²

or, a²/2 = c²

or, a/√2 = c ......(4)

from equations (3) and (4),

a : b : c = a : (√3 + 1)/2 a : a/√2

= 2 : (√3 + 1) : √2