Vector Magnitude: Formula, Examples, and Calculator

Calculate a vector’s magnitude by taking the square root of the sum of the squares of its components. Enter the vector’s coordinates below to get its length, independent of its direction.

Formula

For a three-dimensional vector with components x, y, and z, its magnitude is |v| = √(x² + y² + z²). The components are signed coordinates along the vector’s axes; the magnitude is their nonnegative length. The calculator’s input components are dimensionless numbers by default. If they represent a physical quantity, such as meters or feet, the magnitude has that same unit.

For a vector in two dimensions, use |v| = √(x² + y²). For n dimensions, square every component, add the squares, and take the square root: |v| = √(v₁² + v₂² + … + vₙ²). This is the Euclidean norm, the familiar straight-line length.

Worked Example

Using the calculator’s default values, x = 3, y = 4, and z = 0:

1. Square each component: 3² = 9, 4² = 16, and 0² = 0.
2. Add the squares: 9 + 16 + 0 = 25.
3. Take the square root: √25 = 5.

The vector’s magnitude is 5. If the components are measured in meters, the result is 5 meters; if they are coordinates without physical units, it is simply 5 units.

What Magnitude Means

A vector describes both size and direction. Its components specify how far it extends along each coordinate axis, including the sign that indicates direction. Magnitude describes only its size: it is the vector’s distance from the origin, or equivalently its length when drawn as an arrow.

For example, a displacement of (3, 4) means 3 units in one axis direction and 4 in the other. Its magnitude is 5 units, by the Pythagorean theorem. The same length results for (−3, 4), because squaring removes the sign. These vectors point in different directions but have equal magnitude.

Magnitude cannot be negative. A zero vector, whose components are all zero, has magnitude zero and no defined direction. For a nonzero vector, dividing each component by its magnitude creates a unit vector: one with magnitude 1 that points in the same direction.

How to Calculate It in Different Dimensions

The calculation extends directly to any number of perpendicular coordinate axes. In two dimensions, it is the hypotenuse of a right triangle. In three dimensions, it can be understood by first finding the length in one plane and then combining that length with the third component.

• Two dimensions: (6, 8) has magnitude √(36 + 64) = 10.
• Three dimensions: (2, −3, 6) has magnitude √(4 + 9 + 36) = 7.
• Four dimensions: (1, 2, 2, 4) has magnitude √(1 + 4 + 4 + 16) = 5.

The number of components determines how many squared terms appear, not a different underlying rule. In applications such as physics or data analysis, make sure the components refer to compatible axes and use compatible units before combining them.

When to Use Magnitude and How to Read It

Use magnitude when you need a vector’s overall size but not its direction. Common examples include finding the length of a displacement, the speed represented by a velocity vector, or the size of a force or another component-based quantity. The mathematical operation is the same in each case, but the result’s interpretation and units come from the original vector.

If a vector has components in meters, its magnitude is in meters. If velocity components are in meters per second, the magnitude is in meters per second. Do not combine components expressed in different units without first converting them to a consistent unit. Likewise, the result is a straight-line length in the coordinate system; it is not necessarily the distance traveled along a curved path.

Magnitude is also distinct from a vector’s components. For (3, 4), neither component alone is the magnitude: the length is 5. It is also distinct from the sum of components, which would be 7 here and does not generally measure vector length.

Common Mistakes and Limits

• Forgetting to square: Add the squares of components, not their raw values.
• Keeping component signs in the final result: Negative components are valid, but their squares are nonnegative, and the square root gives a nonnegative magnitude.
• Mixing units: Convert all components to the same unit before calculating. For example, convert feet to meters if other components are in meters.
• Rounding too early: Keep the full component values through the calculation, then round the final magnitude to the precision you need.
• Assuming every “length” is Euclidean: The formula here applies to ordinary perpendicular Cartesian axes. Other mathematical settings can define different norms or distance measures.

For very large or very small numbers, directly squaring components can cause numerical overflow or underflow in some software. A numerically stable implementation rescales the components before squaring. For ordinary hand calculations and typical coordinate values, the standard formula is appropriate.

Frequently Asked Questions

What is the formula for the magnitude of a vector?
For components v₁ through vₙ, the Euclidean magnitude is √(v₁² + v₂² + … + vₙ²). In two dimensions, that is √(x² + y²); in three, it is √(x² + y² + z²).

Can a vector magnitude be negative?
No. Magnitude is a length, so it is zero or positive. Component values can be negative, but squaring them makes their contribution to the magnitude nonnegative.

How do I find the magnitude of a 2D vector?
Square its x and y components, add those squares, and take the square root. For (6, 8), the result is √(6² + 8²) = 10.

Is magnitude the same as the vector’s direction?
No. Magnitude gives size or length, while direction says which way the vector points. Different vectors can have the same magnitude but different directions.

What is the magnitude of the zero vector?
It is 0, because every component is zero and the square root of their squared sum is zero. The zero vector has no defined direction.

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in x | x component | - | 3 | 
in y | y component | - | 4 | 
in z | z component | - | 0 | 
out magnitude | Vector magnitude | - | sqrt(x^2 + y^2 + z^2) | 2