1. The problem is to resolve the vector \( \mathbf{B} \) into its components without using kinematics formulas. 2. To resolve a vector into components, we use trigonometry based on the angle the vector makes with the axes. 3. Suppose \( \mathbf{B} \) makes an angle \( \theta \) with the horizontal axis. 4. The horizontal component \( B_x \) is given by \( B_x = B \cos \theta \). 5. The vertical component \( B_y \) is given by \( B_y = B \sin \theta \). 6. These formulas come from the definitions of cosine and sine in a right triangle formed by the vector and its components. 7. Therefore, the vector \( \mathbf{B} \) can be expressed as \( \mathbf{B} = B_x \hat{i} + B_y \hat{j} = B \cos \theta \hat{i} + B \sin \theta \hat{j} \). 8. This method does not require kinematics formulas, only basic trigonometry. 9. Final answer: \( B_x = B \cos \theta \) and \( B_y = B \sin \theta \).