Adding displacement vectors involves combining them to find the resulting total or net displacement. This resulting vector represents the single overall change in position from the starting point to the ending point after undergoing all the individual displacements.
Understanding Vector Addition
To add two displacement vectors, you combine their effects. This is represented mathematically by the displacement vector formula: V → = V 1 → + V 2 →. This formula shows that the net displacement vector (V →) is the sum of the individual displacement vectors (V 1 → and V 2 →).
This concept extends to adding more than two vectors: V → = V 1 → + V 2 → + V 3 → + ....
How to Add Using Components (Coordinates)
One of the most common and straightforward ways to add displacement vectors, especially in physics and engineering, is by adding their corresponding components or coordinates.
As stated in the reference, to find each coordinate of the net displacement vector, simply add the corresponding coordinate for each displacement vector.
Here's how it works for vectors in 2D or 3D space:
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For 2D Vectors: If you have two displacement vectors, V 1 → = (x₁, y₁) and V 2 → = (x₂, y₂), their sum, the net displacement vector V → = (x, y), is found by adding the x-coordinates and the y-coordinates separately:
- x = x₁ + x₂
- y = y₁ + y₂
- So, V → = (x₁ + x₂, y₁ + y₂)
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For 3D Vectors: If you have V 1 → = (x₁, y₁, z₁) and V 2 → = (x₂, y₂, z₂), their sum V → = (x, y, z) is found by adding each corresponding coordinate:
- x = x₁ + x₂
- y = y₁ + y₂
- z = z₁ + z₂
- So, V → = (x₁ + x₂, y₁ + y₂, z₁ + z₂)
Example: Adding Two 2D Displacement Vectors
Let's say you have two displacements:
- V 1 →: A displacement of 3 units east and 2 units north. In coordinate form, this could be V 1 → = (3, 2).
- V 2 →: A displacement of 1 unit east and 4 units north. In coordinate form, this could be V 2 → = (1, 4).
To find the net displacement V → = V 1 → + V 2 →, we add the corresponding coordinates:
- Add the x-coordinates: 3 + 1 = 4
- Add the y-coordinates: 2 + 4 = 6
So, the net displacement vector is V → = (4, 6). This means the overall change in position is 4 units east and 6 units north from the starting point.
Summary of Adding Vectors by Components
| Vector | x-component | y-component | z-component (if applicable) |
|---|---|---|---|
| V 1 → | x₁ | y₁ | z₁ |
| V 2 → | x₂ | y₂ | z₂ |
| Net V → | x₁ + x₂ | y₁ + y₂ | z₁ + z₂ |
Adding vectors by components is the standard method for calculating the exact sum of displacement vectors. It simplifies the process, especially for vectors that are not aligned with the axes.
