3D Vector Magnitude Calculator

What is 3D Vector Magnitude?

Imagine you're a superhero who can fly in any direction: up, down, left, right, forward, or backward. The path you take is like a 3D vector! The magnitude of a 3D vector is like measuring how far you flew in a straight line, no matter which direction you went. It's the total distance of your flight!

How to Calculate 3D Vector Magnitude

To find the magnitude of a 3D vector, we use a special math trick. We take each part of our flight (how far we went left/right, forward/backward, and up/down), square these numbers, add them up, and then find the square root. It's like adding up all the energy from each direction of your flight!

Formula

If we have a 3D vector \(\vec{v} = (x, y, z)\), its magnitude \(|\vec{v}|\) is:

\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]

Where:

• \(x\) is how far the vector goes left or right
• \(y\) is how far the vector goes forward or backward
• \(z\) is how far the vector goes up or down

Calculation Steps

1. Square each component: \(x^2\), \(y^2\), \(z^2\)
2. Add these squared values: \(x^2 + y^2 + z^2\)
3. Find the square root of this sum

Example

Let's calculate the magnitude of a superhero flight: \(\vec{v} = (3, 4, 5)\)

1. Square each component: \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\)
2. Add these values: \(9 + 16 + 25 = 50\)
3. Find the square root: \(\sqrt{50} \approx 7.07\)

So, our superhero flew about 7.07 units in total!

Visual Representation