As shown, a system of cables supports a bucket. (Figure 2) The dimensions in the

Question

As shown, a system of cables supports a bucket. (Figure 2) The dimensions in the figure are as follows: x1 = 4.40 ft , x2 = 1.80 ft , y1 = 1.20 ft , y2 = 4.15 ft , z1 = 3.20 ft , and z2 = 3.25 ft . If the bucket and its contents have a combined weight of W = 27.0 lb , determine TA, TB, and TC, the tensions in cable segments DA, DB, and DC, respectively. Express your answers numerically in pounds to three significant figures separated by commas.  https://media.cheggcdn.com/media%2F98a%2F98aa650f-ad41-4951-94b7-e7781d9efc3f%2FphpeI8xE3.png

Explanation / Answer

vector DA = (x1 - x2, -y1, z1) = (4.4-1.8, -1.2, 3.2) = (2.6,-1.2,3.2)

unit vector uA   = (2.6,-1.2,3.2)/ 4.29418 = (0.60547,-0.279448,0.745194)

vector DB = (0, -y1, 0) = (0, -1.2, 0),

unit vector uB = (0, -1, 0)

vector DC = (-x2, y2 - y1, z2) = (-1.8, 2.95, 3.25),

unit vector uC =   (-0.379431,0.621846,0.685084)

net force (vector sum) = TAuA + TBuB + Tcuc + 27(0, 0, -1) = 0

0.60547 TA - 0.379431 Tc = 0

-0.279448 TA - TB + 0.621846 Tc = 0

0.745194TA + 0.685084 Tc - 27 = 0

solve them and get

TA = 14.6867 lb, TB = 10.4694 lb, Tc = 23.436 b

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