Vector Addition Calculator

How to Calculate Vector Addition

Vector addition is a fundamental operation in vector algebra. It's used to combine two or more vectors to find a resultant vector. Here's a comprehensive guide on how to perform vector addition:

Vector Addition Formula

For two vectors in 2D space \(\vec{a} = (a_x, a_y)\) and \(\vec{b} = (b_x, b_y)\), their sum \(\vec{c}\) is given by:

\[ \vec{c} = \vec{a} + \vec{b} = (a_x + b_x, a_y + b_y) \]

For two vectors in 3D space \(\vec{a} = (a_x, a_y, a_z)\) and \(\vec{b} = (b_x, b_y, b_z)\), their sum \(\vec{c}\) is given by:

\[ \vec{c} = \vec{a} + \vec{b} = (a_x + b_x, a_y + b_y, a_z + b_z) \]

Where:

• \(\vec{c}\) is the resultant vector
• \(\vec{a}\) and \(\vec{b}\) are the vectors being added
• \(a_x, a_y, a_z\) are the components of vector \(\vec{a}\)
• \(b_x, b_y, b_z\) are the components of vector \(\vec{b}\)

Calculation Steps

1. Identify the components of each vector.
2. Add the x-components of the vectors.
3. Add the y-components of the vectors.
4. If working in 3D, add the z-components of the vectors.
5. Combine the results into a new vector.

Example Calculation

Let's add vectors \(\vec{a} = (3, 4)\) and \(\vec{b} = (1, -2)\) in 2D:

1. Identify components:
   • \(\vec{a} = (3, 4)\)
   • \(\vec{b} = (1, -2)\)
2. Add x-components: \(3 + 1 = 4\)
3. Add y-components: \(4 + (-2) = 2\)
4. Combine results: \(\vec{c} = (4, 2)\)

Therefore, the sum of \(\vec{a}\) and \(\vec{b}\) is \(\vec{c} = (4, 2)\).

Visual Representation

This diagram illustrates the addition of vectors \(\vec{a}\) (blue) and \(\vec{b}\) (green) to form the resultant vector \(\vec{c}\) (red) in 2D space. The resultant vector is the diagonal of the parallelogram formed by the two original vectors.