91Ó°ÊÓ

When dealing with forces acting in various directions, it is essential to understand how to add vectors. Vectors are not like regular numbers; they have both magnitude and direction. To add them, we consider their components individually. Each force acting on an object can be broken down into its respective components using unit vectors

• \(\hat{\mathrm{i}}\) - Representing the x-component.
• \(\hat{\mathrm{j}}\) - Representing the y-component.
• \(\hat{\mathrm{k}}\) - Representing the z-component.

When we add forces, we sum the components separately. This means that for any vectors \( \vec{A} \) and \( \vec{B} \), the resulting vector \( \vec{A} + \vec{B} \) has components:

• \(A_x + B_x\) for the x-direction,
• \(A_y + B_y\) for the y-direction,
• \(A_z + B_z\) for the z-direction.

In the exercise, the vectors \(\vec{F}_1\) and \(\vec{F}_2\) are added by first finding their negative, as required by the equation to solve for the third force.

The net force on an object is the vector sum of all the forces acting upon it. According to Newton's First Law, if an object moves with a constant velocity, the net force is zero. This means all the forces acting on it are balanced, resulting in no change in motion. To maintain a state of motion or rest, the forces must align in such a manner that their sum cancels out.
In the exercise, understanding that the particle moves with constant velocity leads to setting the net force equation to zero:

• \(\vec{F}_1 + \vec{F}_2 + \vec{F}_3 = 0\)

Solving for \(\vec{F}_3\) involves ensuring the sum of the x, y, and z components collectively cancel out. This embodies the definition of a net force being zero by resolving each directional component.

Constant velocity in physics implies that an object is moving in a straight line at a uniform speed. There is no acceleration. In terms of forces, achieving constant velocity requires a balance of all forces, meaning the net force acting is zero.
Newton's First Law explains that an object will move at a constant velocity unless acted upon by an external force. If a particle's speed and direction remain unaltered, it signifies no unbalanced forces are acting on it. In the given exercise, the particle maintains this constant velocity, revealing all exerted forces, including the third unknown force, come together to create equilibrium. The components of these forces precisely balance, maintaining the particle's state of steady motion.