المتجهات في الفضاء: حل المشكلات

A 1000-newton chandelier is proposed for an exhibition hall, supported by three cables fixed to ceiling anchor points. The goal is to find the tension in each cable that keeps the chandelier in equilibrium.

The setup uses a 3D coordinate system, with the chandelier at the origin, and each anchor point assigned coordinates.

The position vectors from the origin to each anchor point give the cable directions. Dividing each position vector by its magnitude gives a unit vector without changing its direction. Since tension acts along the cable direction, each unit vector is then multiplied by an unknown scalar to give the corresponding tension vector.

The chandelier’s weight acts vertically downward and is shown as a force vector.

Since the chandelier remains stationary, it is in static equilibrium. So, the net force is zero, and the tension vectors balance the weight.

This gives a system of three equations by balancing the forces in the x, y, and z directions. Solving this system step by step gives the values of the scalar multipliers. Substituting these into the tension vectors gives the tension in each cable needed to hold the chandelier in place.