The problem asks to determine the three forces acting on the plate that are equivalent to the resultant force $R = 210 \, \text{kN}$. The three forces $T_1$, $T_2$, and $T_3$ act in different directions, and their vectors can be expressed in terms of their components along the $x$, $y$, and $z$-axes. Here's a step-by-step outline of the solution approach:

Step 1: Identify Coordinates of Points

From the diagram:

• The plate is in the $xy$-plane.
• $T_1$ acts from a point located at (1, 0, 0) and directed upwards.
• $T_2$ acts from a point at (3, 0, 0) in a diagonal direction.
• $T_3$ acts from (2, 2, 0) in another diagonal direction.

The forces' directions can be defined by their position vectors relative to the plate and the anchor point at the top, 6 m above the plate in the $z$-direction.

Step 2: Write the Force Vectors

Let each of the force vectors $T_1$, $T_2$, and $T_3$ have components in the $x$, $y$, and $z$ directions, denoted as:

• $\mathbf{T}_1 = (T_{1x}, T_{1y}, T_{1z})$
• $\mathbf{T}_2 = (T_{2x}, T_{2y}, T_{2z})$
• $\mathbf{T}_3 = (T_{3x}, T_{3y}, T_{3z})$

Step 3: Apply Equilibrium Equations

The resultant force $R = 210 \, \text{kN}$ is distributed among the three forces, and the system is in equilibrium. Therefore, the sum of the components of $T_1$, $T_2$, and $T_3$ must equal the resultant force $R$ in all directions.

1. $\sum F_x = 0$
2. $\sum F_y = 0$
3. $\sum F_z = R$

Using the geometry of the situation and the coordinates of the forces, each of these equations can be expressed in terms of the known distances.

Step 4: Solve for Magnitudes of the Forces

Using the equilibrium equations and the geometric relationships (based on the distances given in the figure), solve for the magnitudes of the forces $T_1$, $T_2$, and $T_3$.

Would you like me to go into detail on solving the components of each force vector using the given geometry and apply the equilibrium equations?