Final answer:
In problems involving multiple forces, we can express each force in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force using vector addition and trigonometry.
Step-by-step explanation:
In problems involving multiple forces acting in different directions, we can express each force in Cartesian vector form using the unit vectors i, j, and k. The i vector represents the x-axis, the j vector represents the y-axis, and the k vector represents the z-axis if the problem is in three dimensions.
To find the magnitude and coordinate direction angles of the resultant force, we can use vector addition. We resolve each force into its x, y, and z components, add the corresponding components together, and then use the Pythagorean theorem to find the magnitude of the resultant force. The angle between the resultant force and each coordinate axis can be found using trigonometry.
For example, if we have force F₁ = 3i + 2j + 4k and force F₂ = -2i + 5j - 3k, we can add their respective components (3-2)i + (2+5)j + (4-3)k = i + 7j + k. The magnitude of the resultant force is sqrt(1^2 + 7^2 + 1^2) = sqrt(51), and the coordinate direction angles are atan(1/7), atan(1/sqrt(51)), and atan(1).