3D Vector Subtraction Calculator

What is 3D Vector Subtraction?

Imagine you're a space explorer with a magical spaceship. You can fly in any direction: up/down, left/right, and forward/backward. Each journey is like a 3D vector! Now, if you want to find out how to get from one planet to another, you need to subtract one vector from another. That's what 3D vector subtraction is all about!

How to Calculate 3D Vector Subtraction

To subtract 3D vectors, we simply subtract the matching parts of each vector. It's like playing a game where you take away pieces from one stack and compare it to another. We subtract the x-parts, the y-parts, and the z-parts. 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 difference \(\vec{c}\) 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. Subtract the x-components: \(c_x = a_x - b_x\)
2. Subtract the y-components: \(c_y = a_y - b_y\)
3. Subtract 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 subtract two space journeys: \(\vec{a} = (5, 3, 2)\) and \(\vec{b} = (2, 1, 4)\)

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

So, to get from planet B to planet A, our space explorer needs to go 3 units right, 2 units forward, and 2 units down!

Visual Representation