3D Normalize Vector â Calculator

What is a 3D Normalized Vector?

Imagine you have a magic arrow that can point in any direction in space. A normalized vector is like making that arrow always the same length, no matter which way it points. It's like having a standard-sized magic wand that can show any direction!

How to Calculate a 3D Normalized Vector

To normalize a 3D vector, we make it have a length of exactly 1 unit. It's like shrinking or stretching our magic arrow to always be 1 unit long. We do this by dividing each part of the vector by its magnitude (its original length).

Formula

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

\[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} = \left(\frac{x}{\sqrt{x^2 + y^2 + z^2}}, \frac{y}{\sqrt{x^2 + y^2 + z^2}}, \frac{z}{\sqrt{x^2 + y^2 + z^2}}\right) \]

Where:

• \(x\), \(y\), and \(z\) are the original vector components
• \(|\vec{v}|\) is the magnitude (length) of the original vector
• \(\hat{v}\) is the normalized vector (always has a magnitude of 1)

Calculation Steps

1. Calculate the magnitude of the original vector: \(|\vec{v}| = \sqrt{x^2 + y^2 + z^2}\)
2. Divide each component by the magnitude: \(\hat{x} = \frac{x}{|\vec{v}|}\), \(\hat{y} = \frac{y}{|\vec{v}|}\), \(\hat{z} = \frac{z}{|\vec{v}|}\)
3. The normalized vector is \(\hat{v} = (\hat{x}, \hat{y}, \hat{z})\)

Example

Let's normalize the vector \(\vec{v} = (3, 4, 5)\)

1. Calculate the magnitude: \(|\vec{v}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50} \approx 7.07\)
2. Divide each component by 7.07:
   \(\hat{x} = \frac{3}{7.07} \approx 0.42\)
   \(\hat{y} = \frac{4}{7.07} \approx 0.57\)
   \(\hat{z} = \frac{5}{7.07} \approx 0.71\)
3. The normalized vector is \(\hat{v} \approx (0.42, 0.57, 0.71)\)

Visual Representation