What is the magnitude of a vector?

It is the length of the vector: each component squared, the squares added, and the square root of that total. It is one number, and it says how big the vector is without saying anything about which way it points. A vector drawn from the origin to a point has the same magnitude as the distance from the origin to that point, which is why the expression looks like the distance formula — it is the distance formula.

How do I work out the length of a vector by hand?

Square every component, add the squares together, and take the square root of the total. For 3 4 that is 9 plus 16, which is 25, and the root of 25 is 5. For 1 1 it is 1 plus 1, which is 2, and the root of 2 is 1.4142 to four places. The middle step is the only one with any room for error, and it is the step the page's reference table lists for every row.

Can I enter a two-dimensional or a three-dimensional vector?

Either, and there is no setting to change: two entries is a two-dimensional vector and three entries is a three-dimensional one. The same box takes both because the calculation is the same calculation — a vector in the plane is a vector in space whose third component happens to be 0, and its length is the same square root with two terms under it instead of three.

What is the magnitude of the zero vector?

Zero, and this page reports it rather than refusing the input. Every component is 0, so every square is 0, so the sum is 0 and its root is 0. Nothing in the calculation divided by the length, which is the reason there is no error here — the unit vector page refuses this same input precisely because there the length is a divisor.

Does the magnitude tell me the direction?

No, and it cannot: it is a single number, and a direction needs at least two. The vectors 3 4, −3 4 and 0 5 have three different directions and all three are five long, so the length cannot be used to tell them apart. If direction is what you need, read the components directly or normalise the vector into a unit vector, which keeps the direction and discards the size instead.

What is a vector magnitude used for?

Comparing sizes, and feeding the next step of a longer calculation. Two forces or two velocities are ranked by magnitude when the directions are beside the point. It is also the number that a great many other formulas divide by or multiply with — normalising a vector, projecting one vector onto another, and the dot product all consume it — so working it out on its own is common even when the length is not the final answer.

Vector Magnitude Calculator

Result

5.0000

Magnitude of the vector

A vector magnitude calculator gives the length of a vector from its components. Type two or three numbers separated by spaces or commas — 3 4, or 2 3 6 for a three-dimensional vector — and the page squares each component, adds the squares and takes the square root. That one number is the magnitude of a vector, and it is also the distance from the origin to the point those components name, since the length of an arrow drawn from the origin is exactly that distance. Two components and three components are both accepted, and the count of entries is what says which you meant. Being a single number is both the use and the limit of the answer: it is how sizes get compared when directions are irrelevant, and it is exactly what is thrown away when you need to know which way something points. Three quite different vectors share a length of 5 in the table below. Nothing here divides by anything, so the zero vector is an ordinary input and its length of 0 is an ordinary answer.

Vectors and their magnitudes

Vector	Magnitude of the vector
3 4	5
5 12	13
1 1	1.4142
-3 4	5
2 3 6	7
1 1 1	1.7321
0 5	5
0 0	0

Eight vectors and their lengths. The second column is the one to read first, because three of the eight rows print the same value: 3 4, −3 4 and 0 5 are three different vectors — one up and to the right, one up and to the left, one straight up — and every one of them is 5 long. That repetition is the point of the table. A magnitude is a single number and a single number cannot say which way a vector points, so the page that reports only the magnitude has thrown that information away by design. The first two rows are the whole-number cases, 3-4-5 and 5-12-13, where the sum of the squares happens to be a perfect square and nothing is rounded. The third and sixth rows are the equal-component cases, in the plane and in space, and both give roots that never terminate. The fifth row is the three-dimensional whole-number case: 2-3-6 gives exactly 7. The seventh row lies along an axis, so its length is just the component that is not zero. The eighth row is the zero vector, printing 0 — which this page answers rather than refuses, because a length is not a divisor here. Column one holds the input exactly as typed; the spaces in it are separators and not part of the numbers.

Formula

|u| ＝ √（u₁² ＋ u₂² ＋ … ＋ uₙ²）

Vector u: The vector whose length is wanted: two or three numbers separated by spaces or commas. Nothing else marks which kind it is — the number of entries is what says whether you meant a vector in the plane or one in space
|u|: The length itself: every component squared, the squares added, and the square root of that total. This is the number on the panel
u₁² ＋ u₂² ＋ …: The sum of the squares, which the worked examples below list term by term. It is the only intermediate step in the calculation, and seeing it written out is what makes the result checkable by hand
√: The square root, which is why so many answers are irrational. 3 4 gives 5 exactly because 9 plus 16 happens to be 25; 1 1 gives 1.4142 and no finite decimal is equal to it
Two components: A vector in the plane. Both counts are accepted and neither is a special case: a two-dimensional vector is a three-dimensional one whose third component is 0
Zero vector: A vector whose components are all 0. Its length is 0, and that is an answer rather than a refusal — this page divides by nothing, which is precisely why the zero vector is accepted here and turned away by the pages that normalise
Four decimal places: How wide the answer is written. Exact when the sum of the squares is a perfect square, rounded to four places otherwise
No unit: Nothing on this page knows what the components were measured in, and the answer carries whatever unit they did. A length of 5 means 5 of whatever was typed

Use this page when how big something is matters and which way it points does not. The clearest case is comparing sizes: two forces, two velocities or two displacements are ranked by length alone, and the direction only gets in the way. The second case is a length that is one step in a longer calculation — the magnitude is the divisor in normalising a vector, the denominator in the projection formula and one of the two factors in a dot product, so it is worked out on its own far more often than it is used for its own sake. The third case is the most concrete of all: the components of a vector are the coordinates of a point, so its length is how far that point is from the origin, and the expression is the distance formula you may already know under a different name. What the answer does not carry is direction, and that is worth being explicit about because it is easy to over-read. A length of 5 came from 3 4, from −3 4 and from 0 5 alike. If you need to know which of those you have, the length is not the page you want — the components or the unit vector are. A two-component input is fine and is not padded: it is a vector in the flat plane, and its length is the same distance formula with two terms. The zero vector is accepted, and its length of 0 is a true statement rather than an error.

Worked examples

3 4
1. Square each component: 3² ＝ 9 and 4² ＝ 16
2. Add the squares: 9 ＋ 16 ＝ 25
3. Take the square root: √25 ＝ 5
The input the page loads with and the cleanest case there is. 9 plus 16 is 25, a perfect square, so the answer is a whole number with nothing rounded off — which happens because 3 and 4 are two legs of the 3-4-5 right triangle. Check it the other way round: a point three across and four up is five from the origin, which is the same thing as saying the vector 3 4 is five long.
5 12
1. Square each component: 5² ＝ 25 and 12² ＝ 144
2. Add the squares: 25 ＋ 144 ＝ 169
3. Take the square root: √169 ＝ 13
The next Pythagorean triple after 3-4-5, and the one that shows an exact answer is not a one-off. Both components are larger than in the first example and the answer still comes out whole. It also shows that the answer is not bounded by either component: 13 is bigger than 5 and bigger than 12, which is what the square root of a sum does.
1 1
1. Square each component: 1² ＝ 1 and 1² ＝ 1
2. Add the squares: 1 ＋ 1 ＝ 2
3. Take the square root: √2 ＝ 1.414214, printed as 1.4142
The ordinary case rather than the exception: most sums of squares are not perfect squares, so most answers are irrational and get rounded to four places. √2 is the simplest of them and the one worth recognising — it turns up whenever a unit is split evenly between two components. Check the definition holds backwards: 1.4142 squared is 1.99996, which is 2 within the rounding.
0 0
1. Square each component: 0² ＝ 0 and 0² ＝ 0
2. Add the squares: 0 ＋ 0 ＝ 0
3. Take the square root: √0 ＝ 0
The zero vector, and this page answers it rather than turning it away. The length of a vector from the origin to the origin is 0, and nothing in the calculation divided by anything, so there is no reason to refuse. The pages that normalise a vector do refuse this same input, because there the length is a divisor — the verdict belongs to the question being asked, not to the input.

Limitations

Two or three components are accepted, and only those: a single number is a signed quantity rather than a direction, and four or more do not correspond to a length anyone asked this page for. A two-component input is not padded to three — it lies in the flat plane, and the same direction written in three coordinates has a third component of 0, which is why the box does not need to ask which kind you meant. Components are read as ordinary numbers: a comma followed by a space separates components, a comma with no space after it is read as a decimal point (1,5 is one and a half), and a thousands grouping is refused rather than guessed at — write 1500, not 1,500. The four decimal places are a display width and not a claim of precision, and the answer is computed from the unrounded sum of squares, so the printed figure may not square back to exactly the number you started with. The length is a pure number: nothing here knows whether the components were metres, newtons or nothing at all, and a magnitude carries whatever unit they did. The one thing the answer genuinely cannot do is tell you anything about direction. A length of 5 is shared by 3 4, by −3 4 and by 0 5, and a page that reports only the length has, by construction, thrown the distinction away. For direction use the components themselves or the unit vector built from them; for a comparison that ignores direction this page is the whole answer, which is the point of it.

Frequently asked questions

What is the magnitude of a vector?: It is the length of the vector: each component squared, the squares added, and the square root of that total. It is one number, and it says how big the vector is without saying anything about which way it points. A vector drawn from the origin to a point has the same magnitude as the distance from the origin to that point, which is why the expression looks like the distance formula — it is the distance formula.
How do I work out the length of a vector by hand?: Square every component, add the squares together, and take the square root of the total. For 3 4 that is 9 plus 16, which is 25, and the root of 25 is 5. For 1 1 it is 1 plus 1, which is 2, and the root of 2 is 1.4142 to four places. The middle step is the only one with any room for error, and it is the step the page's reference table lists for every row.
Can I enter a two-dimensional or a three-dimensional vector?: Either, and there is no setting to change: two entries is a two-dimensional vector and three entries is a three-dimensional one. The same box takes both because the calculation is the same calculation — a vector in the plane is a vector in space whose third component happens to be 0, and its length is the same square root with two terms under it instead of three.
What is the magnitude of the zero vector?: Zero, and this page reports it rather than refusing the input. Every component is 0, so every square is 0, so the sum is 0 and its root is 0. Nothing in the calculation divided by the length, which is the reason there is no error here — the unit vector page refuses this same input precisely because there the length is a divisor.
Does the magnitude tell me the direction?: No, and it cannot: it is a single number, and a direction needs at least two. The vectors 3 4, −3 4 and 0 5 have three different directions and all three are five long, so the length cannot be used to tell them apart. If direction is what you need, read the components directly or normalise the vector into a unit vector, which keeps the direction and discards the size instead.
What is a vector magnitude used for?: Comparing sizes, and feeding the next step of a longer calculation. Two forces or two velocities are ranked by magnitude when the directions are beside the point. It is also the number that a great many other formulas divide by or multiply with — normalising a vector, projecting one vector onto another, and the dot product all consume it — so working it out on its own is common even when the length is not the final answer.

References

Related calculators

Vector CalculatorVector magnitude, direction and components.Vector Addition CalculatorAdd or subtract vectors, and find the resultant.Vector Projection CalculatorProjection of one vector onto another, scalar and vector.