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in Laws of motion by (52.4k points)

The climber A is being helped up an icy slope by two friends. His mass is 80 kg, and the direction cosines of the force exerted on him by the slope are cos θx = -0.286, cos θy = 0.429, cos θz = 0.857. The y axis is vertical. If the climber is in equilibrium in the position shown, what are the tensions in the ropes AB and AC and the magnitude of the force exerted on him by the slope?

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Get the unit vectors parallel to the ropes using the coordinates of the end points. Express the tensions in terms of these unit vectors, and solve the equilibrium conditions. The rope tensions, the normal force, and the weight act on the climber. The coordinates of points A, B, C are given by the problem, A(3, 0, 4), B(2, 2, 0), C(5, 2, -1).

The vector locations of the points A, B, C are:

The unit vector parallel to the tension acting between the points A, B in the direction of B is

where the last was given by the problem statement. The forces are expressed in terms of the unit vectors,

Substitute and collect like terms,

We have three linear equations in three unknowns. The solution is:

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