If a unit vector v makes equal acute angles with the vectors p = i - 2 j + 2 k , q = 2 i + j - 2 k and r = 3 i + 4 j , then v is:

Step-by-step solution

Let v = x i + y j + z k . Since v makes equal acute angles with p , q , and r , we have: = v p | p | = v q | q | = v r | r | > 0 The magnitudes of the given vectors are: | p | = 1^2 + (-2)^2 + 2^2 = 3 | q | = 2^2 + 1^2 + (-2)^2 = 3 | r | = 3^2 + 4^2 + 0^2 = 5 Equating the expressions for gives: x - 2y + 2z 3 = 2x + y - 2z 3 = 3x + 4y 5 From the first two parts: x - 2y + 2z = 2x + y - 2z x = 4z - 3y Substituting x into the first and third parts: (4z - 3y) - 2y + 2z 3 = 3(4z - 3y) + 4y 5 6z - 5y 3 = 12z - 5y 5 5(6z -