Vector Magnitude Calculator

About Vector Magnitude Calculator

The magnitude of a vector is its length. In 2D, it's the Pythagorean theorem; in 3D, add the third component squared.

How to use this calculator

1. Enter the vector components, such as \(v_1\), \(v_2\), and \(v_3\).
2. Make sure each component is written with the correct sign, positive or negative.
3. Click calculate to compute the vector’s length.
4. Check the result, which is the magnitude \(|\vec{v}|\).

The formula explained

The formula \(|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\) computes the length of a 3D vector by squaring each component, adding them, and taking the square root. For 2D vectors, the unused third component is treated as 0.

• \(\vec{v}\) = the vector whose length you want to find
• \(v_1\) = the first component of the vector
• \(v_2\) = the second component of the vector
• \(v_3\) = the third component of the vector
• \(|\vec{v}|\) = the magnitude, or length, of the vector

Step by step method

1. Write the components of the vector.
2. Square each component.
3. Add the squared values and take the square root.

Worked example

Problem. Find the magnitude of the vector \(\vec{v} = \langle 3, 4, 12 \rangle\).

1. Square each component: \(3^2 = 9\), \(4^2 = 16\), and \(12^2 = 144\).
2. Add the squares: \(9 + 16 + 144 = 169\).
3. Take the square root: \(|\vec{v}| = \sqrt{169} = 13\).

Tips and common mistakes

• Remember to square negative components too, for example \((-5)^2 = 25\).
• If your vector has only 2 components, use \(v_3 = 0\) so the formula still works.

Frequently asked questions