Vector Magnitude Calculator
Find a vector's length by entering its components; this calculator computes the square root of sum of squared components rapidly.
Inputs
Actions
Equation Preview
Results
Magnitude:
—Steps:
Helping Notes
Notes: The magnitude is always non-negative. For 3D, z component is included. If all components are zero, result is zero.
Inputs accept integer and decimal values. Use the example buttons to quickly fill the fields and compute.
What is Vector Magnitude Calculator
A Vector Magnitude Calculator is a simple tool that calculates the length (magnitude) of a vector from its components or endpoints. Vectors describe direction and size. The magnitude is a scalar representing how long the vector is — an essential value in physics, computer graphics, engineering, and robotics where movement, force, and spatial measurements matter.
About the Vector Magnitude Calculator
This calculator accepts either component form (for example, <em>(x, y, z)</em>) or two endpoint coordinates (for example, point A and point B). It applies the Euclidean distance formula to return the scalar length. The tool is tolerant of 2D and 3D inputs, and can be extended to n-dimensions by using the same root-sum-of-squares approach. The formulas are shown in renderable math markup so they scale responsively with the page’s math-rendering library.
How to Use this Vector Magnitude Calculator
1. Enter components (e.g., x, y, z) or enter two points A(x1, y1, z1) and B(x2, y2, z2).
2. Choose whether you are providing components (vector form) or endpoints (points form).
3. Click Calculate to see the magnitude. For components, the calculator computes the square root of the sum of squared components.
For endpoints, it computes the distance between the two points and presents the same scalar length.
Formulas (render-ready)
For a vector with components \( \langle v_x, v_y \rangle \) in 2D:
\( \; \| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2} \; \)
For a vector with components \( \langle v_x, v_y, v_z \rangle \) in 3D:
\( \; \| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2 + v_z^2} \; \)
Given two points \( A(x_1,y_1,z_1) \) and \( B(x_2,y_2,z_2) \), the magnitude of \( \overrightarrow{AB} \) is:
\( \; \| \overrightarrow{AB} \| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \; \)
Examples of Vector Magnitude Calculator
Example 1 — 2D components: vector \( \langle 3, 4 \rangle \).
Formula: \( \| \mathbf{v} \| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5 \).
Example 2 — 3D components: vector \( \langle -2, 1, 2 \rangle \).
Formula: \( \| \mathbf{v} \| = \sqrt{(-2)^2 + 1^2 + 2^2} = \sqrt{4+1+4} = \sqrt{9} = 3 \).
Example 3 — Endpoints: A(1,2,3) and B(4,6,7).
Differences: \( \Delta x = 3, \Delta y = 4, \Delta z = 4 \).
Magnitude: \( \sqrt{3^2 + 4^2 + 4^2} = \sqrt{9 + 16 + 16} = \sqrt{41} \approx 6.4031 \).
Responsive formula notes
The formulas above are provided in math markup that will scale and reflow when rendered by a math rendering library on the page, making them responsive across screen sizes. Keep the formula blocks inline or in display mode (centered) to preserve readability.
What is the magnitude of a vector?
The magnitude of a vector is its length — a non-negative scalar representing how long the vector is in space.
How do I compute magnitude from components?
Square each component, sum them, then take the square root: \( \| \mathbf{v} \| = \sqrt{\sum v_i^2} \).
Can the calculator handle 2D and 3D vectors?
Yes — the same root-sum-of-squares method works for 2D, 3D, and higher dimensions by including all components.
What input formats are supported?
Typical inputs are component form (x,y,z) or endpoint pairs A and B; both produce the same scalar magnitude.
Is the magnitude ever negative?
No — magnitude is always zero or positive. A zero vector has magnitude 0.
How is magnitude used in physics?
It measures quantities like speed (magnitude of velocity) or force size (magnitude of force vector).
Does direction matter for magnitude?
No — magnitude ignores direction and returns only the vector’s length.
How do I get magnitude from polar or spherical coordinates?
Convert the coordinates to Cartesian components first, then apply the root-sum-of-squares formula.
Can I compute magnitude manually?
Yes — use the formulas shown above on paper or in a calculator by squaring components, summing, and taking a square root.
What precision should I expect?
Precision depends on input precision; calculators typically return a decimal rounded to a chosen number of places.
How to handle negative components?
Negative components are squared, so sign does not affect the final magnitude result.
Is magnitude the same as distance?
Yes for vectors: magnitude is the distance from the origin to the vector tip; distance between two points uses the same formula on their difference.