determine the required values of the tension in ac and ad so that the resultant of the three forces applied at A is verticalFigure:A wire is coupled from three sides

Fac = 21 KN

Fad = 64.29 KN

Explanation:

Step 1: Find the unit vectors in BA, CA, and DA directions

BA = - 16 i + 48 j - 12 k

CA = - 16 i + 48 j + 24 k

DA = 14 i + 48 j

The following magnitudes

mag (BA) = sqrt (16 ^2 + 48^2 + 12^2) = 52

mag (CA) = sqrt (16 ^2 + 48^2 + 24^2) = 56

mag (DA) = sqrt (14 ^2 + 48^2) = 50

Now compute unit vectors

unit (BA) = (-4 / 13) i + (12 / 13) j - (3 / 13) k

unit (CA) = (-2 / 7) i + (6 / 7) j - (3 / 7) k

unit (DA) = (7 / 25) i + (24 / 25) j

Step 2: Compute Force Vectors F (ml) dot unit (ML)

Fab = (-4*Fab / 13) i + (12*Fab / 13) j - (3*Fab / 13) k

Fac = (-2*Fac / 7) i + (6*Fac / 7) j - (3*Fac / 7) k

Fad = (7*Fad / 25) i + (24*Fad / 25) j

Step 3: Equilibrium force equations in x and z direction

Fr, x = 0 = 39 * (-4 / 13) - Fac * (2 / 7) + Fad * (7 / 25)

Fr, z = 0 = 39 * (-3 / 13) + Fad * (3 / 7)

Step 4: Solve the above equations for Fca and Fda

Fca = 21 KN

Fda = 64.29 KN