To find the components of the vertical force in the directions parallel to and normal to the plane, we need to resolve the force vector into its components. Let's assume the magnitude of the vertical force is F and the angle between the force and the positive x-axis is θ.
The component of the force parallel to the plane is F_parallel = F * sin(θ). This is because sin(θ) is the ratio of the opposite side (the component parallel to the plane) to the hypotenuse (the magnitude of the force).
The component of the force normal to the plane is F_normal = F * cos(θ). This is because cos(θ) is the ratio of the adjacent side (the component normal to the plane) to the hypotenuse.
To show that the total force is the sum of the two component forces, we can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle, the square of the hypotenuse (the magnitude of the force) is equal to the sum of the squares of the other two sides (the components of the force).
So, we have:
F^2 = F_parallel^2 + F_normal^2
Substituting the expressions for F_parallel and F_normal, we get:
F^2 = (F * sin(θ))^2 + (F * cos(θ))^2
Simplifying, we get:
F^2 = F^2 * sin^2(θ) + F^2 * cos^2(θ)
F^2 = F^2 * (sin^2(θ) + cos^2(θ))
F = F
Therefore, we have shown that the total force is indeed the sum of the two component forces.