2D Normalize Vector Calculator

What is a 2D Normalized Vector?

Imagine you have a toy car that can drive in any direction on a flat table. A normalized vector is like giving your car a standard amount of fuel, so it always drives exactly one unit of distance, no matter which direction you point it. It keeps the direction of your original vector but changes its length to exactly 1.

How to Calculate a 2D Normalized Vector

To normalize a 2D vector, we follow these simple steps:

• Find out how long our original vector is (its magnitude)
• Divide each part of our vector (x and y) by this length

Formula

For a vector \(\vec{v} = (x, y)\), its normalized form \(\hat{v}\) is:

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

Where:

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

Calculation Steps

1. Write down the x and y components of your vector
2. Calculate the magnitude: \(|\vec{v}| = \sqrt{x^2 + y^2}\)
3. Divide x by the magnitude: \(\frac{x}{|\vec{v}|}\)
4. Divide y by the magnitude: \(\frac{y}{|\vec{v}|}\)
5. Your normalized vector is \(\hat{v} = (\frac{x}{|\vec{v}|}, \frac{y}{|\vec{v}|})\)

Example

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

1. We have \(\vec{v} = (3, 4)\)
2. Calculate magnitude: \(|\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
3. Normalize x: \(\frac{3}{5} = 0.6\)
4. Normalize y: \(\frac{4}{5} = 0.8\)
5. Our normalized vector is \(\hat{v} = (0.6, 0.8)\)

So, the normalized form of \(\vec{v}\) is (0.6, 0.8).

Visual Representation