Question

Vectors →A and →B are two orthogonal vectors in the xy-plane, and they have identical magnitudes. If →A=3.0ˆi+4.0ˆj, find →B.

a) →B: (4.0, -3.0)
b) →B: (-4.0, 3.0)
c) →B: (-3.0, -4.0)
d) →B: (3.0, -4.0)

Answer 1

Vector →B, which is orthogonal to vector →A = 3.0ˆi + 4.0ˆj and has the same magnitude, is (4.0, -3.0).

Step-by-step explanation:

The question involves finding vector →B given that vectors →A and →B are orthogonal vectors in the xy-plane with identical magnitudes. The vector →A is provided as 3.0ˆi + 4.0ˆj. To be orthogonal, the dot product of →A and →B must be zero, and since they have identical magnitudes, the components of →B must be either the negative or positive permutation of the components of →A. So, →B can either be (4.0, -3.0) or (-4.0, 3.0).

To find vector →B, we can use the fact that →A and →B are orthogonal vectors and have identical magnitudes. Since →B is orthogonal to →A, it will have the same magnitude but its components will be switched and have opposite signs. Therefore, if →A = 3.0ˆi + 4.0ˆj, →B will be -4.0ˆi + 3.0ˆj. Therefore, the correct answer is b) →B: (-4.0, 3.0).

However, since the positive i component corresponds to the positive j component in →A, the i component of →B should correspond to the negative of →A's j component and vice versa for the j component of →B. Hence, the correct vector →B that is orthogonal to →A and has the same magnitude is given by (4.0, -3.0).