If vector a and vector b make an angle cos⁻¹(5/9) with each other, then |a + b| = √2 |a - b| for |a| = n|b|. The integer value of n i s _______.
Physics · Mathematical tools and vector
JEE Mains 2024 09 Apr Morning Shift

Question

If vector a and vector b make an angle cos⁻¹(5/9) with each other, then |a + b| = √2 |a - b| for |a| = n|b|. The integer value of n i s _______.

G1 Physics · Accepted Answer

Given: cosθ = 5/9 and: |a + b| = √2 |a - b| Squaring: a² + b² + 2ab cosθ = 2[a² + b² - 2ab cosθ] Therefore: 6ab cosθ = a² + b² Using: cosθ = 5/9 (10/3)ab = a² + b² Given: a = nb Substitute: (10/3)nb² = n²b² + b² Therefore: 3n² - 10n + 3 = 0 (n - 3)(3n - 1) = 0 So: n = 3 or 1/3 The integer value is: n = 3