To find the resultant vector in component form, you add the corresponding components of the individual vectors.
Finding the resultant vector, often denoted as R, when you have two or more vectors in component form (like A = ⟨Aₓ, Aᵧ⟩ and B = ⟨Bₓ, Bᵧ⟩) is a straightforward process based on vector addition. The resultant vector represents the single vector that has the same effect as the original vectors combined.
Based on the provided reference, "We can add them to get the resultant vector in component form. So the resultant vector is going to be the sum of f1 and f2 what we're going to do is we're going to add the x components together". This principle extends to all components (x, y, z, etc.).
The Process
Here's how to calculate the resultant vector R = ⟨Rₓ, Rᵧ⟩ for two vectors A = ⟨Aₓ, Aᵧ⟩ and B = ⟨Bₓ, Bᵧ⟩:
- Add the x-components: The x-component of the resultant vector (Rₓ) is the sum of the x-components of the individual vectors (Aₓ + Bₓ).
- Add the y-components: The y-component of the resultant vector (Rᵧ) is the sum of the y-components of the individual vectors (Aᵧ + Bᵧ).
- Combine the results: The resultant vector in component form is ⟨Rₓ, Rᵧ⟩ = ⟨Aₓ + Bₓ, Aᵧ + Bᵧ⟩.
This method is easily extendable to three dimensions (adding z-components) or any number of vectors.
Step-by-Step Example
Let's find the resultant vector R for two vectors:
- V₁ = ⟨3, 5⟩
- V₂ = ⟨-1, 2⟩
Using the component addition method:
- Rₓ = V₁ₓ + V₂ₓ = 3 + (-1) = 2
- Rᵧ = V₁ᵧ + V₂ᵧ = 5 + 2 = 7
So, the resultant vector R in component form is ⟨2, 7⟩.
Understanding the Components
Each component represents the projection of the vector onto the respective axis. By adding the x-components, you find the total displacement or force along the x-axis. Similarly, adding the y-components gives the total effect along the y-axis. Combining these sums gives the single resultant vector that describes the overall effect.
| Vector | X-component | Y-component |
|---|---|---|
| V₁ | 3 | 5 |
| V₂ | -1 | 2 |
| Resultant R | 3 + (-1) = 2 | 5 + 2 = 7 |
This component form is particularly useful for calculations and manipulations of vectors in coordinate systems.
