Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system.
The Correct Answer and Explanation is:
To determine the magnitude and coordinate direction angles of the resultant force, we first need the vector components of all forces involved. Since you haven’t provided a specific diagram or force vectors, I’ll explain the general method with a common example and then show how to sketch it.
Example:
Suppose we are given three forces acting at a point in space:
- F₁ = 100 N along the x-axis → F₁ = ⟨100, 0, 0⟩
- F₂ = 80 N in the y-direction → F₂ = ⟨0, 80, 0⟩
- F₃ = 60 N in the z-direction → F₃ = ⟨0, 0, 60⟩
Step 1: Add the Force Components
The resultant force FR is the sum of the vectors: FR=F1+F2+F3=⟨100,0,0⟩+⟨0,80,0⟩+⟨0,0,60⟩=⟨100,80,60⟩FR = F₁ + F₂ + F₃ = ⟨100, 0, 0⟩ + ⟨0, 80, 0⟩ + ⟨0, 0, 60⟩ = ⟨100, 80, 60⟩FR=F1+F2+F3=⟨100,0,0⟩+⟨0,80,0⟩+⟨0,0,60⟩=⟨100,80,60⟩
Step 2: Find the Magnitude of the Resultant
Use the 3D magnitude formula: ∣FR∣=(100)2+(80)2+(60)2=10000+6400+3600=20000=141.42 N|FR| = \sqrt{(100)^2 + (80)^2 + (60)^2} = \sqrt{10000 + 6400 + 3600} = \sqrt{20000} = 141.42 \, \text{N}∣FR∣=(100)2+(80)2+(60)2=10000+6400+3600=20000=141.42N
Step 3: Coordinate Direction Angles
Use cosine relations: cos(α)=FRx∣FR∣=100141.42=0.707⇒α=cos−1(0.707)=45∘\cos(\alpha) = \frac{FR_x}{|FR|} = \frac{100}{141.42} = 0.707 \Rightarrow \alpha = \cos^{-1}(0.707) = 45^\circcos(α)=∣FR∣FRx=141.42100=0.707⇒α=cos−1(0.707)=45∘ cos(β)=FRy∣FR∣=80141.42=0.566⇒β=cos−1(0.566)=55.1∘\cos(\beta) = \frac{FR_y}{|FR|} = \frac{80}{141.42} = 0.566 \Rightarrow \beta = \cos^{-1}(0.566) = 55.1^\circcos(β)=∣FR∣FRy=141.4280=0.566⇒β=cos−1(0.566)=55.1∘ cos(γ)=FRz∣FR∣=60141.42=0.424⇒γ=cos−1(0.424)=64.9∘\cos(\gamma) = \frac{FR_z}{|FR|} = \frac{60}{141.42} = 0.424 \Rightarrow \gamma = \cos^{-1}(0.424) = 64.9^\circcos(γ)=∣FR∣FRz=141.4260=0.424⇒γ=cos−1(0.424)=64.9∘
Sketching the Vector
Draw a 3D coordinate system with x, y, and z axes. From the origin, draw a vector pointing into the first octant with components 100 in x, 80 in y, and 60 in z. This vector represents FR.
Summary
- Resultant Force Magnitude: 141.42 N
- Direction Angles:
- α = 45° with x-axis
- β = 55.1° with y-axis
- γ = 64.9° with z-axis
