If a vector 3 i + 4 j - 5 k is rotated through a certain angle about the origin in the anti-clockwise direction, then the components of the new vector are a + 1, -3, 5 . The possible values of a is

Step-by-step solution

When a vector is rotated about the origin, its magnitude remains unchanged. Let the initial vector be v ₁ = 3 i + 4 j - 5 k . Magnitude of v ₁ is | v ₁| = 3^2 + 4^2 + (-5)^2 = 9 + 16 + 25 = 50 . Let the new vector be v ₂ = (a+1) i - 3 j + 5 k . Magnitude of v ₂ is | v ₂| = (a+1)^2 + (-3)^2 + 5^2 = (a+1)^2 + 9 + 25 = (a+1)^2 + 34 . Since | v ₁| = | v ₂| , we have: 50 = (a+1)^2 + 34 Squaring both sides: 50 = (a+1)^2 + 34 (a+1)^2 = 16 a+1 = 4 a = 3 or a = -5 Answer: -5 or 3