Two vectors a i +b j + k and 2 i -3 j +4 k are perpendicular to each other. When 3 a+2 b=7 , the ratio of a to b is x 2 . The value of x is

Step-by-step solution

Given vectors V₁ = a i + b j + k and V₂ = 2 i - 3 j + 4 k are perpendicular, so their dot product equals zero: 2a - 3b + 4 = 0 This simplifies to 2a - 3b = -4 The second condition gives 3a + 2b = 7 Solving this system by elimination, multiply the first equation by 2 and the second by 3: 4a - 6b = -8 9a + 6b = 21 Adding eliminates b and yields 13a = 13 , so a = 1 Substituting into 3a + 2b = 7 gives 3 + 2b = 7 , hence b = 2 The ratio a b = 1 2 equals x 2 , so x = 1 Answer: C