Vector Magnitude Calculator

What the magnitude of a vector measures

A vector carries two pieces of information: how much, and which way. The magnitude is the how much part, a single non-negative number that says how long the arrow is. A velocity of 3 m/s east and a velocity of 3 m/s north have the same magnitude and different directions, and the magnitude of a velocity is what a speedometer shows.

In two or three dimensions the magnitude is the straight-line length from the tail of the vector to its tip. It does not depend on where the vector is drawn, only on its components, so the vector from (0, 0) to (3, 4) and the vector from (10, 10) to (13, 14) are the same vector and both have magnitude 5.

The NASA Glenn Research Center describes vectors as arrows with length and direction and notes that the length is found from the components, which is the rule used here. The same convention runs through the vector chapter of Paul Dawkins' calculus notes at Lamar University, where magnitude is written with two vertical bars around the vector name.

Magnitude is never negative. A component can be, a direction angle can be, but a length cannot. If a calculation returns a negative magnitude, the sign belongs to a component or a direction, and the magnitude should be taken as the positive root.

The formula, component by component

The rule is Pythagoras applied once per dimension. For a vector with components x, y and z, square each component, add the squares, then take the square root of the total. In two dimensions the z term drops out and the formula reduces to the familiar right-triangle result.

The order of the steps matters for reading the result. Squaring first removes the signs, which is why (-3, 4) and (3, 4) share a magnitude of 5. Adding first and squaring afterwards gives a different and wrong answer: a vector of (2, 3, 6) would come out at 11 instead of 7.

The last row is the one people get wrong by hand. The minus sign disappears in the square, so the answer is 13 and not 7, which is what 5 minus 12 would suggest.

Worked example: the vector (2, 3, 6)

Take a vector that points 2 units along x, 3 along y and 6 along z. Square each: 4, 9 and 36. The sum is 49, and the square root of 49 is exactly 7, so the arrow is 7 units long.

The same answer comes out of the component view. The point (2, 3, 0) sits 3.6056 units from the origin in the flat plane, since 4 + 9 = 13. Rising 6 units from there makes a right triangle with legs 3.6056 and 6, and the hypotenuse works out to 7. Handling the third dimension in one step, rather than two, is the reason the formula squares all three components at once.

Scaling shows the relationship between components and length. Doubling every component to (4, 6, 12) gives 16 + 36 + 144 = 196, and the square root of 196 is 14, exactly twice the original. Halving gives (1, 1.5, 3), whose magnitude is 3.5. A vector scaled by a factor k has a magnitude scaled by the absolute value of k, so reversing a vector to (-2, -3, -6) leaves the length at 7.

A table makes the same point across a few vectors, and the pattern is that length grows more slowly than the components. Going from (2, 3, 6) to (4, 6, 12) doubles each component but only doubles the length, while adding a modest 1 to the largest component moves the length by less than 0.1.

Two dimensions and the distance between two points

The most common use of a magnitude is not a vector drawn from the origin but the gap between two points. Subtract the coordinates, and the difference is a vector whose magnitude is the distance. The points (1, 2) and (4, 6) differ by (3, 4), so the distance between them is 5.

This is the same calculation the Pythagorean theorem calculator performs for a right triangle, with the two legs being the horizontal and vertical separations. In three dimensions the same subtraction applies to the z coordinates and the distance picks up the third square.

Because the components are squared, the order of subtraction does not matter: (1, 2) to (4, 6) and (4, 6) to (1, 2) both give 5. That property is why distance is a metric and direction is not.

The unit vector: same direction, length one

Dividing each component by the magnitude produces a vector of length 1 that points the same way. For (3, 4, 0) the magnitude is 5, so the unit vector is (0.6, 0.8, 0). For (2, 3, 6) it is (0.2857, 0.4286, 0.8571), and the squares of those three numbers add back to 1.000.

A unit vector is the standard way to separate direction from size. In graphics it is the normal that sets a surface angle, in physics it is the direction of a force with the magnitude factored out, and in machine learning it is the direction used in cosine similarity. The dot product calculator relies on the same normalization, since the cosine of the angle between two vectors is their dot product divided by both magnitudes.

The one case with no answer is the zero vector. (0, 0, 0) has magnitude 0, and dividing by zero is undefined, so there is no direction to report. A calculation that returns a unit vector of NaN or Infinity has hit this case, and the fix is to check the magnitude before dividing.

Direction angles and direction cosines

Once the magnitude is known, each component can be turned into an angle. The direction cosine for the x axis is x divided by the magnitude, and the angle is the inverse cosine of that value. The three angles are usually labelled alpha, beta and gamma, and they run from 0 to 180 degrees.

For (2, 3, 6) the direction cosines are 0.2857, 0.4286 and 0.8571. The inverse cosines are 73.40, 64.62 and 31.00 degrees. The largest component gives the smallest angle, which is the sanity check to apply: the vector leans most toward the z axis because 6 is the biggest number.

The three cosines are not independent. Their squares always sum to 1 for any non-zero vector, as the table above shows, which means that knowing two of the angles fixes the third. A vector lying along an axis has one angle of 0 degrees and two of 90 degrees, and a vector on the diagonal (1, 1, 1) has three equal angles of 54.74 degrees.

Two dimensions use a single angle instead. That angle is the inverse tangent of y over x, and the trigonometry calculator covers the quadrant corrections, since inverse tangent alone only returns values between -90 and 90 degrees.

Mistakes that change the answer

The first mistake is adding the components instead of the squares. For (2, 3, 6) the sum of the components is 11, while the length is 7. The triangle inequality explains the gap: the straight path is never longer than the route along the axes, and it is shorter whenever more than one component is non-zero.

The second is dropping a component. Leaving out the z value of (2, 3, 6) gives the square root of 13, which is 3.6056, less than half the right answer of 7. In three dimensions all three terms are needed even when one component looks small, because a component of 1 still adds 1 to the sum.

The third is a unit mix. Adding inches to millimetres, or mixing a component in metres with one in centimetres, produces a number that has no physical meaning. Convert to one unit before squaring, and note that the result carries that unit, so a vector in metres has a magnitude in metres.

The fourth is rounding too early. Rounding 0.4286 to 0.43 before squaring changes the length check in the third decimal place, and in a chain of rotations those small errors accumulate. Keep full precision through the calculation and round only the number that gets displayed.

A fifth error is expecting a negative magnitude to mean a reversed vector. Direction lives in the components, not in the length, so a vector pointing backwards has negative components and a positive magnitude.

Where magnitude shows up, and which page to use

Speed is the magnitude of velocity, and that is why a car going around a curve at a steady 30 mph has a changing velocity and a constant speed. Force vectors use the same rule: a 3 N push along x and a 4 N push along y combine to a 5 N push at 53.13 degrees. In statistics, the magnitude of a deviation vector is the quantity that standard deviation scales down, which is why the standard deviation calculator and this page share the same sum of squares.

The vector addition calculator answers the neighbouring question: it takes vectors as magnitudes and angles, converts them to components, and adds two or three of them. This page starts where that one ends, taking the components as input and reporting the length, the unit vector and the direction angles.

Use this page when the components are known, for example from a dataset with x, y and z columns, a CAD model, or a sensor that reports three axes. Use the addition page when the inputs arrive as compass bearings and speeds. Both pages report direction, but only the addition page produces a new vector from several.

One practical check before trusting any of these numbers: the magnitude must be at least as large as the biggest single component and no larger than the sum of the absolute values. For (2, 3, 6) that means a value between 6 and 11, and 7 falls inside the window.