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Given that vec(A)+vec(B)+vec(C )=vec(0)....

Given that `vec(A)+vec(B)+vec(C )=vec(0)`. Out of three vectors,two are equal in magnitude and the magnitude of the third vectors is `sqrt(2)` times that of either of the two having equal magnitude. Find the angles between the vectors.

Text Solution

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Given

`A=B,C=sqrt(2A)=sqrt(2B)`.
From (figure) `alpha=beta` and `alpha+beta+gamma=180^(@)`
`rArr gamma=180^(@)-2alpha`
Apply Lami's theorem: `A/(sin alpha)=C/(sin gamma)`
`rArrA/(sin alpha)=sqrt(2A)/(sin alpha (180-2alpha))`
`rArr1/(sin alpha)=sqrt(2)/(sin2 alpha)`
`1/(sin alpha)=sqrt(2)/(2sin alphacos alpha)rArrcos alpha=1/sqrt(2)rArralpha=45^(@)`
`rArrbeta=45^(@)` and `gamma=180^(@)-2 alpha=90^(@)`
Angle between `vec(A)` and `vec(B)=180^(@)-gamma=90^(@)`,
Angle between `vec(B)` and `vec(C )=180-alpha=135^(@)`,
Angle between `vec(C )` and `vec(A)=180^(@)-beta=135^(@)`.
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