Adding vectors in vector form is a fundamental operation in mathematics and physics, representing the combined effect or resultant of multiple vector quantities. When vectors are expressed using their components, the process is remarkably simple.
To add two vectors in vector form, you add their corresponding components. This means combining the 'x' components, 'y' components, and 'z' components (if applicable) of the vectors separately.
The Core Principle: Adding Components
As stated in the reference, the key to adding vectors in component form is straightforward:
"To add two vectors, we simply add their components. In other words, add the x component of the first vector to the x component of the second and so on for y and z. The answers you get from adding the x, y, and z components of your original vectors are the x, y, and z components of your new vector."
This principle applies whether your vectors are in two dimensions (2D) or three dimensions (3D).
Step-by-Step Process
Here's a simple breakdown of how to add vectors when they are given in their component form:
- Identify the Components: Note the individual components of each vector. A 2D vector $\vec{A}$ might be written as $\langle A_x, A_y \rangle$, and a 3D vector as $\langle A_x, A_y, A_z \rangle$.
- Group Corresponding Components: Pair the x-components together, the y-components together, and the z-components together from the vectors you are adding.
- Perform the Addition: Add the numbers within each group.
- Add the x-components to get the new x-component.
- Add the y-components to get the new y-component.
- Add the z-components to get the new z-component (if applicable).
- Form the Resultant Vector: The sums you calculated in Step 3 are the components of the new vector, often called the resultant vector.
Example: Adding 2D Vectors
Let's illustrate with an example of adding two 2D vectors, $\vec{A}$ and $\vec{B}$.
Suppose:
$\vec{A} = \langle 2, 3 \rangle$
$\vec{B} = \langle 4, -1 \rangle$
To find the sum $\vec{R} = \vec{A} + \vec{B}$, we add the corresponding components:
- New x-component: $2 + 4 = 6$
- New y-component: $3 + (-1) = 2$
So, the resultant vector is $\vec{R} = \langle 6, 2 \rangle$.
This can be easily visualized in a table:
| Vector | x-component | y-component |
|---|---|---|
| $\vec{A}$ | 2 | 3 |
| $\vec{B}$ | 4 | -1 |
| $\vec{A} + \vec{B}$ | $2+4=6$ | $3+(-1)=2$ |
Thus, $\vec{A} + \vec{B} = \langle 6, 2 \rangle$.
Extending to 3D
The process is identical for 3D vectors. If you have $\vec{C} = \langle C_x, C_y, C_z \rangle$ and $\vec{D} = \langle D_x, D_y, D_z \rangle$, their sum $\vec{S} = \vec{C} + \vec{D}$ is simply:
$\vec{S} = \langle C_x + D_x, C_y + D_y, C_z + D_z \rangle$
Why Add Components?
Adding vectors component-wise works because each component of a vector represents its magnitude along a specific axis in the coordinate system (like the x-axis, y-axis, etc.). When you add vectors, you are essentially combining their effects along each independent direction. Adding the components separately ensures that you are correctly accounting for the combined influence along each axis.
This method is efficient and forms the basis for other vector operations, such as vector subtraction (which is just adding the negative vector, achieved by negating its components).
Vector addition in component form is a powerful tool for solving problems in physics (like calculating resultant forces or velocities) and engineering.
