Question

The resultant of two vectors A and B acting at a point is equal to √3B & makes angle 30° with the direction of A. Prove that A=B or A=2B.

Answer 1

$\huge{\underline{\underline{\mathfrak{\sf{Solution}}}}}$

❥...agnitude of resultant of A and B is given by,

R = √(A² + B² + 2ABcosФ)

❥..According to question ,

Magnitude of resultant = half of Magnitude of B

e.g., √(A² + B² + 2A.BcosФ) = B/2

Taking square both sides,

A² + B² + 2A.BcosФ = B²/4

A² + 2ABcosФ + 3B²/4 = 0 --------(1)

❥..Also, A and R is perpendicular upon

each other ,

e.g., A.(A + B) = 0

A.A + A.B = 0

A² + A.BcosФ = 0

cosФ = - A/B , put it in equation (1)

A² + 2A.B(-A/B) + 3B²/4 = 0

A² - 2A² + 3B²/4 = 0

A = √3B/2

❥..Now, cosФ = -A/B = -√3B/2B = -√3/2

cosФ = cos150° ⇒Ф = 150°

Hence angle between A and B = 150°

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