3-d Rectangular components Direction Cosines Rectangular Components in Space r • The vector F is contained in the plane OBAC. r • Resolve F into horizontal and vertical components. Fy = F cosθ y Fh = F sin θ y • Resolve F h into rectangular components Fx = Fh cos φ = F sin θ y cos φ Fy = Fh sin φ = F sin θ y sin φ Rectangular Components in Space Sample Problem 2.7 SOLUTION: • Based on the relative locations of the points A and B, determine the unit vector pointing from A towards B. Direction of the force is defined by the location of two points, M ( x1 , y1 , z1 ) and N ( x2 , y 2 , z 2 ) • Apply the unit vector to determine the components of the force acting on A. The tension in the guy wire is 2500 N. Determine: • Noting that the components of the unit vector are the direction cosines for the vector, calculate the corresponding angles. a) components Fx, Fy, Fz of the force acting on the bolt at A, b) the angles θx, θy, θz defining the direction of the force 1 Sample Problem 2.7 SOLUTION: • Determine the unit vector pointing from A towards B. r r r AB = (− 40 m ) i + (80 m ) j + (30 m )k AB = (− 40 m )2 + (80 m )2 + (30 m )2 = 94.3 m r − 40 ⎞ r ⎛ 80 ⎞ r ⎛ 30 ⎞ r ⎟k ⎟j +⎜ ⎟i + ⎜ ⎝ 94.3 ⎠ ⎝ 94.3 ⎠ ⎝ 94.3 ⎠ r r r = −0.424 i + 0.848 j + 0.318k λ = ⎛⎜ Sample Problem 2.7 • Noting that the components of the unit vector are the direction cosines for the vector, calculate the corresponding angles. r r r r λ = cos θ x i + cos θ y j + cos θ z k r r r = −0.424 i + 0.848 j + 0.318k θ x = 115.1o θ y = 32.0o θ z = 71.5o • Determine the components of the force. r r F = Fλ r r r = (2500 N )(− 0.424 i + 0.848 j + 0.318k ) r r r = (− 1060 N )i + (2120 N ) j + (795 N )k 3-D Equilibrium of Particles r ∑F Ext =0 -Equilibrium equation -Can be used to find unknown forces -A vector equation equivalent to 3 scalar component equations. Example. Cable AB is attached to the top of the vertical 3-m post, and its tension is 50 kN. What are the tensions in cables AO, AC, and AD. 2