Vector Length Calculator
Every vector has a length, the single number that says how big it is once direction is set aside: a force, a velocity, a displacement, reduced to its size. This calculator takes the x, y and optional z components of one vector and returns that length, also called its magnitude or norm, from the square root of the squared components, rounded to six decimal places.
Blank components count as 0. If all three components are zero or blank, the calculator asks for at least one component value instead of printing 0.
Quick Answer
The length of the vector (3, 4, 12) is exactly 13: √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13. Those components are pre-filled above, so one click on Calculate reproduces it. Leave the z field blank and the same formula runs in 2D, where (3, 4) has length 5.
Key takeaways
- Length, magnitude and norm are three names for the same number: the square root of the summed squared components.
- Signs never matter. Squaring erases them, so (-3, -4, 0) and (3, 4, 0) both have length 5.
- A blank component counts as 0, which is what makes the same three fields serve both 2D and 3D vectors.
- The result carries the unit of the components: metres in, metres out. It is never negative.
- Quick Answer
- How to use the vector length calculator
- The formula: Pythagoras, extended one axis at a time
- Worked example: the length of (3, 4, 12)
- Reading the result: what the length does and does not say
- What competing calculators miss
- Three mistakes that corrupt a length calculation
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the vector length calculator
Three fields, one of them optional, and the answer stays hidden until you calculate. The whole interaction takes seconds.
- Enter the x and y components of your vector. Decimals and negative values are accepted; the pre-filled example is (3, 4, 12).
- Add z only for a 3D vector. For a flat 2D vector, leave z blank or at 0 and it simply drops out of the formula.
- Click Calculate (or press Enter). The length appears as the headline number and in the Vector Length field, ready to copy or share.
- Reset restores the (3, 4, 12) example and hides the result.
Two behaviours are worth knowing before they surprise you. A blank or non-numeric field counts as 0 rather than producing an error, so a mistyped field quietly changes the answer instead of stopping it: check the components before you trust the number. And if all three components are zero or blank, an alert asks you to “enter at least one component value” and no result is printed. The zero vector does have length 0 mathematically, but this tool treats an all-zero form as an unfilled form, a guard against reporting 0 for a vector you never typed.
The formula: Pythagoras, extended one axis at a time
The length formula is the Pythagorean theorem with one extra term per dimension. In 2D the components are the legs of a right triangle and the length is its hypotenuse; in 3D the same trick runs twice.
3D length: |v| = √(x² + y² + z²)
Between two points: use the component differences first: √((x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²)
OpenStax University Physics defines the magnitude of a vector in exactly these terms in its scalars and vectors section: the length of the arrow, computed from the components once the vector is resolved along the axes, as its components section works through. Two relatives of this formula are worth recognising. The length of a resultant after vectors are combined is the same computation applied to the summed components, which is what the vector addition calculator reports as its magnitude. And the length is hiding inside the dot product: a vector dotted with itself gives x² + y² + z², so the dot product calculator run with the same vector twice returns the squared length, 169 for the worked example below.
Worked example: the length of (3, 4, 12)
The pre-filled vector (3, 4, 12) is a classic because its squares land on a perfect square. Nothing about the method changes for messier vectors; only the tidiness of the answer does.
Between two points, the same example does double duty. The segment from (1, 2, 3) to (4, 6, 15) has components (3, 4, 12) after subtracting the endpoints, so its length is also 13. The calculator takes components, not endpoints, so that one subtraction happens by hand, or with the vector addition calculator’s Subtract mode. Six more runs of this calculator make the checks below; type any row in and you get the same figure back.
| Vector | Squared sum | Length | Why it is in the table |
|---|---|---|---|
| (3, 4, 12) | 169 | 13 | The worked example: an exact whole number |
| (3, 4, blank) | 25 | 5 | Blank z counts as 0, the classic 2D case |
| (1, 2, 2) | 9 | 3 | Smallest tidy 3D triple |
| (-3, -4, 0) | 25 | 5 | Signs vanish under the squares |
| (1.5, 2.5, 3.5) | 20.75 | 4.555217 | Decimals: the six-decimal rounding on display |
| (0, 0, -7) | 49 | 7 | A single-axis vector: length is that component’s size |
Reading the result: what the length does and does not say
The length answers one question completely and stays silent on another: it gives the size, and nothing about where the vector points. Four facts frame every result.
The silence on direction is easy to underestimate. The vectors (3, 4), (5, 0) and (-4, 3) all have length 5, and no length calculation can tell them apart. When direction matters, the companion quantity is the unit vector: each component divided by the length. This calculator does not output one (it returns the length only, by design), but the division is quick by hand: for (3, 4, 12) it is (3 ÷ 13, 4 ÷ 13, 12 ÷ 13) = (0.230769, 0.307692, 0.923077), a vector of length 1 pointing the same way. Lengths also feed larger formulas: the magnitude of a cross product is the product of the two lengths times the sine of the angle between them, which the cross product calculator evaluates from components directly.
What competing calculators miss
Magnitude tools are simple, so the differences are in the details:
- Dimension selectors that overshoot (CalculatorsHub, CalculatorLib): 4D and 5D modes headline the feature list, yet most real work is 2D and 3D, and the extra modes push the actual inputs further down the page. This tool keeps three fields and lets a blank z do the dimension switching.
- Answer-only tools (CalculatorDock and similar): a correct number with no account of rounding, no blank-field rule, and no warning that a mistyped field silently counts as 0 in some tools and errors in others. Here both behaviours are stated on the page and next to the result.
- Bundled extras presented as the tool’s own output (magnitude pages that add squared magnitude and unit vectors): useful numbers, but readers walk away unsure which line the calculator actually guarantees. This page keeps the tool to the length alone and shows the unit vector as a hand follow-up, labelled as such.
- Zero-vector evasion: several rivals print 0 for an empty form without comment, teaching the wrong lesson about empty inputs. This tool’s alert forces the distinction between “the zero vector” and “nothing entered yet”.
- The gap this page fills: the exact rounding rule (six decimals, trailing zeros dropped), the blank-equals-zero rule, the all-zero alert explained rather than hidden, and verified runs including negative, decimal and single-axis vectors.
Three mistakes that corrupt a length calculation
Most wrong lengths come from the setup, not the arithmetic. Watch for these three.
- Typing endpoints instead of components. The fields want the vector’s components. For a segment from (1, 2, 3) to (4, 6, 15), subtract first: (3, 4, 12), length 13. Typing (4, 6, 15) directly answers a question nobody asked.
- Mixing units across components. A vector with x in metres and y in centimetres has no meaningful length until both share a unit. The calculator cannot detect the mix; it trusts the fields.
- Stopping at the squared sum. 169 is not the length of (3, 4, 12); it is the length squared. The square root is the step that turns an area-like total back into a distance, and skipping it is the most common hand-calculation slip of all.
Frequently asked questions
Square each component, add the squares, and take the square root: length = √(x² + y² + z²). For the vector (3, 4, 12) that is √(9 + 16 + 144) = √169 = 13. Leave out the z term, or leave the z field blank, for a 2D vector.
Yes. Length, magnitude and norm all name the same quantity: the size of the vector with direction set aside, written |v|. Physics courses usually say magnitude, geometry says length, and linear algebra says norm, but the formula and the number are identical.
No. Components can be negative, but each one is squared before the sum, and a square root is never negative. The vector (-3, -4, 0) has length 5, exactly like (3, 4, 0). The smallest possible length is 0, for the zero vector.
It counts as 0, so the calculator runs the 2D form of the formula on x and y alone. With x = 3, y = 4 and z blank, the result is 5. The same blank rule applies to any field: a blank x or y also counts as 0, which is worth remembering if a result looks too small.
That alert appears when all three components are zero or blank. The zero vector (0, 0, 0) does have length 0, but an all-zero form usually means the numbers were never typed, so the tool asks for a real component instead of printing a misleading 0. Any single non-zero component, however small, calculates normally.
The arithmetic runs at full precision and the display rounds to six decimal places, with trailing zeros dropped. So (1, 1, 1) shows 1.732051, while an exact answer like the length of (3, 4, 12) shows as 13 rather than 13.000000. Six decimals is far beyond what component measurements usually justify.
Subtract the starting point from the ending point to get components, then use those components here. From (1, 2, 3) to (4, 6, 15), the differences are (3, 4, 12) and the length is 13. This calculator takes components rather than endpoints, so the subtraction is a separate step.
No. It returns the length only. To get the unit vector, divide each component by the length: for (3, 4, 12) that is (3 ÷ 13, 4 ÷ 13, 12 ÷ 13) = (0.230769, 0.307692, 0.923077). The unit vector points the same way as the original and has length exactly 1.
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References
- OpenStax, Rice University. University Physics Volume 1, Section 2.1: Scalars and Vectors (magnitude as the length of the vector, the zero vector). Accessed October 24, 2026.
- OpenStax, Rice University. University Physics Volume 1, Section 2.2: Coordinate Systems and Components of a Vector (magnitude rebuilt from components). Accessed October 24, 2026.
- OpenStax, Rice University. Calculus Volume 3, Section 2.2: Vectors in Three Dimensions (three-dimensional components and distance between points). Accessed October 24, 2026.
- Eric W. Weisstein, Wolfram MathWorld. Vector Norm (the Euclidean norm as the square root of summed squares). Accessed October 24, 2026.
- Eric W. Weisstein, Wolfram MathWorld. Pythagorean Theorem (the right-triangle result the 2D formula extends). Accessed October 24, 2026.
- OpenStax, Rice University. Algebra and Trigonometry 2e, Section 10.8: Vectors (magnitude from components in the plane, unit vectors). Accessed October 24, 2026.