Resolving Vectors at an Angle

Resolving a vector means splitting a vector into two component vectors at right angles to each other.T_A = 14.6N T_B = 17.9NThese tensions answer the following problem. A 20 N weight hangs at rest from two strings. String A makes 30° with the horizontal and string B makes 45° with the horizontal. Find the tension in each string.The weight is in equilibrium, so the components of the forces must balance in every direction. Resolve each tension into horizontal and vertical components.Horizontally (left = right):T_A cos 30° = T_B cos 45°T_B = T_A × (0.866 ÷ 0.707) = 1.225 T_AVertically (up = down):T_A sin 30° + T_B sin 45° = 200.500 T_A + (0.707 × 1.225 T_A) = 1.366 T_A = 20T_A = 14.6 N and T_B = 1.225 × 14.6 = 17.9 NThe steeper string, B, carries the larger tension.The same method works for any direction: the component of a force F along a line at angle θ to it is F cosθ. For a weight W on a slope inclined at θ to the horizontal, the component down the slope is W sinθ and the component perpendicular to the slope is W cosθ.