Question

21.) Two vectors A and B have precisely equal magnitudes. If magnitude
A + B to be larger than the magnitude of A - B by a factor n, the ane
between them is
1) 2 tan-(1/n) 2) tan-|(1/n) 3) tan-(1/2n) 4) 2tan" (1/2n)​

Answer 1

Answer:

α = Tan⁻¹(2n/(n² - 1))

Explanation:

Two vectors A and B have precisely equal magnitudes.

Let say Magnitude of A & B = K

let say Angle between them = α

magnitude of A + B = √A² + B² + 2ABCosα

magnitude of A + B = √K² + K² + 2KKCosα

magnitude of A + B = √2K²( 1 + Cosα)

magnitude of A - B = √A² + B² - 2ABCosα

magnitude of A - B = √K² + K² - 2KKCosα

magnitude of A - B = √2K²( 1 - Cosα)

√2K²( 1 + Cosα) = n √2K²( 1 - Cosα)

Squaring both sides

=> 1 + Cosα = n² (1 - Cosα)

=> Cosα (1 + n²) = n² - 1

=> Cosα = (n² - 1)/(n² + 1)

=> Secα = (n² + 1)/ (n² - 1)

Sec²α = Tan²α + 1

=> Tan²α = ((n² + 1)/ (n² - 1))²  - 1

=> Tan²α = ((n² + 1)/ (n² - 1) + 1)((n² + 1)/ (n² - 1) - 1)

=> Tan²α = (2n²)(2)/ (n² - 1)²

=> Tanα = 2n/(n² - 1)

=> α = Tan⁻¹(2n/(n² - 1))