3D Vector Addition Calculator

What is 3D Vector Addition?

Imagine you're a superhero with the power to fly in any direction. When you fly, you can go up/down, left/right, and forward/backward. Each flight is like a 3D vector! Now, if you take two flights one after another, the total distance you've traveled is like adding two 3D vectors together. That's what 3D vector addition is all about!

How to Calculate 3D Vector Addition

To add 3D vectors, we simply add the matching parts of each vector. It's like putting together puzzle pieces that fit perfectly. We add the x-parts together, the y-parts together, and the z-parts together. It's as easy as that!

Formula

If we have two 3D vectors \(\vec{a} = (a_x, a_y, a_z)\) and \(\vec{b} = (b_x, b_y, b_z)\), their sum \(\vec{c}\) is:

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

Where:

• \(a_x\) and \(b_x\) are how far each vector goes left or right
• \(a_y\) and \(b_y\) are how far each vector goes forward or backward
• \(a_z\) and \(b_z\) are how far each vector goes up or down

Calculation Steps

1. Add the x-components: \(c_x = a_x + b_x\)
2. Add the y-components: \(c_y = a_y + b_y\)
3. Add the z-components: \(c_z = a_z + b_z\)
4. Write the result as a new vector: \(\vec{c} = (c_x, c_y, c_z)\)

Example

Let's add two superhero flights: \(\vec{a} = (3, 1, 2)\) and \(\vec{b} = (1, 4, -1)\)

1. Add x-components: \(3 + 1 = 4\)
2. Add y-components: \(1 + 4 = 5\)
3. Add z-components: \(2 + (-1) = 1\)
4. Result: \(\vec{c} = (4, 5, 1)\)

So, after these two flights, our superhero has moved 4 units right, 5 units forward, and 1 unit up!

Visual Representation