You add vectors together by adding their corresponding components.
Understanding Vector Addition
Vectors are mathematical objects that have both magnitude and direction. They are often represented as ordered pairs or triples of numbers (e.g., (x, y) in two dimensions or (x, y, z) in three dimensions). To add vectors, you simply add the corresponding components of each vector.
Component-wise Addition
Suppose you have two vectors, a = (x₁, y₁) and b = (x₂, y₂). The sum of these vectors, denoted as a + b, is calculated as follows:
a + b = (x₁ + x₂, y₁ + y₂)
This means you add the x-components together (x₁ + x₂) and the y-components together (y₁ + y₂).
Example
Let's say a = (2, 4) and b = (1, 5). Then:
a + b = (2 + 1, 4 + 5) = (3, 9)
Therefore, the sum of the vectors (2, 4) and (1, 5) is the vector (3, 9).
Vector Addition in Three Dimensions
The same principle applies to vectors in three dimensions. If a = (x₁, y₁, z₁) and b = (x₂, y₂, z₂), then:
a + b = (x₁ + x₂, y₁ + y₂, z₁ + z₂)
Graphical Representation
Vector addition can also be visualized graphically. One common method is the "tip-to-tail" method:
- Draw the first vector.
- Starting at the tip (arrowhead) of the first vector, draw the second vector.
- The resultant vector (the sum) is the vector drawn from the tail (starting point) of the first vector to the tip of the second vector.
The component-wise addition method and the graphical method always result in the same vector.
Summary
To add vectors, simply add their corresponding components. This applies to vectors in any number of dimensions. You can also visualize vector addition graphically using methods like the tip-to-tail method.
