Two particles A and B are moving with constant velocities. The velocity vector of particle A is given by v _A = 3 i + x j + 4 k (in m/s) and that of particle B is v _B = 4 i + 3 j + y k (in m/s). If both particles are moving with the same speed and their directions of motion are strictly perpendicular to each other, the value of x (given x > 0 ) is _____ .

Step-by-step solution

Since the particles are moving with the same speed, the magnitudes of their velocity vectors are equal: | v _A| = | v _B| 3^2 + x^2 + 4^2 = 4^2 + 3^2 + y^2 25 + x^2 = 25 + y^2 x^2 = y^2 y = x Since the directions of motion are perpendicular, the dot product of their velocity vectors must be zero: v _A v _B = 0 (3)(4) + (x)(3) + (4)(y) = 0 12 + 3x + 4y = 0 Case 1: If y = x 12 + 3x + 4x = 0 7x = -12 x = - 12 7 This is rejected because it is given that x > 0 . Case 2: If y = -x 12 + 3x + 4(-x) = 0 12 - x = 0 x = 12 Th