Unit Vector Calculator (and Vector Magnitude)

This unit vector calculator finds the length (magnitude) of a vector and the unit vector that points the same way. Results are exact where possible: ( |v| ) in simplest radical form and each component of the unit vector as a fraction or radical, with decimals alongside. For 2D and 3D vectors it also gives the direction angles.

It’s free and needs no sign-up.

How to use it

1. Enter the vector’s components separated by commas, for example 3, 4 or 2, -1, 2. Up to six components are accepted.
2. Press Calculate.

Formulas

The magnitude of ( v = langle v_1, v_2, ldots, v_n rangle ) comes from the Pythagorean theorem:

$$|v| = sqrt{v_1^2 + v_2^2 + cdots + v_n^2}$$

The unit vector divides every component by that length:

$$hat{u} = frac{v}{|v|} = leftlangle frac{v_1}{|v|}, frac{v_2}{|v|}, ldots, frac{v_n}{|v|} rightrangle$$

It has length 1 and the same direction as v. Finding it is called normalizing the vector.

Worked example: ⟨3, 4⟩

As loaded above: ( |v| = sqrt{3^2 + 4^2} = sqrt{25} = 5 ), so

$$hat{u} = leftlangle frac{3}{5}, frac{4}{5} rightrangle = langle 0.6, 0.8 rangle$$

Check: ( 0.6^2 + 0.8^2 = 0.36 + 0.64 = 1 ).

Worked example: ⟨2, −1, 2⟩

( |v| = sqrt{4 + 1 + 4} = sqrt{9} = 3 ), so ( hat{u} = leftlangle frac{2}{3}, -frac{1}{3}, frac{2}{3} rightrangle ).

Worked example: ⟨1, 1, 1⟩

( |v| = sqrt{3} ), so each component is ( frac{1}{sqrt{3}} ). Rationalizing the denominator gives

$$hat{u} = leftlangle frac{sqrt{3}}{3}, frac{sqrt{3}}{3}, frac{sqrt{3}}{3} rightrangle approx langle 0.577, 0.577, 0.577 rangle$$

The calculator writes exact answers in this rationalized form.

Direction angles

The angles ( alpha, beta, gamma ) that v makes with the x-, y- and z-axes satisfy ( cosalpha = frac{v_1}{|v|} ), ( cosbeta = frac{v_2}{|v|} ), ( cosgamma = frac{v_3}{|v|} ). In other words, the components of the unit vector are the direction cosines. For ( langle 3, 4 rangle ), ( alpha approx 53.13^circ ) and ( beta approx 36.87^circ ).

Why a unit vector calculator is helpful

• Directional derivatives need a unit direction: ( D_{hat{u}} f = nabla f cdot hat{u} ). Using a non-unit vector scales the answer by mistake (see gradient and directional derivative).
• Physics splits a force into magnitude × direction: ( F = |F|,hat{u} ).
• Graphics and data normalize vectors so only direction matters.

To make a vector of any length L in the same direction, multiply the unit vector by L. For example, a vector of length 10 in the direction of ( langle 3, 4 rangle ) is ( 10 cdot langle 0.6, 0.8 rangle = langle 6, 8 rangle ).

Common mistakes

• Forgetting to square negative components. ( (-1)^2 = 1 ), so negative entries still add to the length.
• Dividing only some components by the length. Every component is divided.
• The zero vector. ( langle 0, 0 rangle ) has no direction and no unit vector.

Step-by-step method

1. Square each component.
2. Add the squares and take the square root: that is ( |v| ).
3. Divide every component by ( |v| ).
4. Simplify or rationalize, and check that the squares of the new components add to 1.

Worked example: ⟨−5, 12⟩

( |v| = sqrt{25 + 144} = sqrt{169} = 13 ), so ( hat{u} = leftlangle -frac{5}{13}, frac{12}{13} rightrangle ). The negative component stays negative: the unit vector points into the same quadrant as v.

Worked example: the direction from one point to another

The unit vector pointing from ( A(1, 2) ) to ( B(4, 6) ): first ( vec{AB} = langle 4 – 1, 6 – 2 rangle = langle 3, 4 rangle ), whose length is 5, so the direction is ( langle 0.6, 0.8 rangle ). Also, the length of ( vec{AB} ) is the distance between the points; see the distance formula.

Worked example: a velocity

A drone flies with velocity ( langle 6, 8, 0 rangle ) m/s. Its speed is ( |v| = 10 ) m/s and its direction is ( langle 0.6, 0.8, 0 rangle ). To fly the same direction at 25 m/s, use ( 25langle 0.6, 0.8, 0 rangle = langle 15, 20, 0 rangle ) m/s. Splitting a vector into speed × direction like this is used throughout physics.

Unit vector notation

Vectors are often written with ( mathbf{i}, mathbf{j}, mathbf{k} ): ( langle 3, 4 rangle = 3mathbf{i} + 4mathbf{j} ). Its unit vector is ( frac{3}{5}mathbf{i} + frac{4}{5}mathbf{j} ). Also, a hat (( hat{u} )) marks a unit vector.

Normal vectors

A unit normal vector is perpendicular to a line or surface and has length 1. The line ( 3x + 4y = 12 ) has normal ( langle 3, 4 rangle ), so its unit normal is ( langle 0.6, 0.8 rangle ). For a plane through three points, find the normal with the cross product and then normalize it here.

Practice problems

1. ( langle 6, 8 rangle ).
2. ( langle 1, 2, 2 rangle ).
3. ( langle 0, -7 rangle ).

Answers: 1. Length 10, unit vector ( langle 0.6, 0.8 rangle ). 2. Length 3, unit vector ( leftlangle frac13, frac23, frac23 rightrangle ). 3. Length 7, unit vector ( langle 0, -1 rangle = -mathbf{j} ).

A unit vector from an angle

In 2D, the unit vector making an angle θ with the positive x-axis is ( langle costheta, sintheta rangle ). For ( theta = 30^circ ) it is ( leftlangle frac{sqrt{3}}{2}, frac12 rightrangle ). This is the point at angle θ on the unit circle, which is why the circle has radius 1.

The direction of a line

The line ( y = 2x + 1 ) rises 2 for every 1 across, so it has direction vector ( langle 1, 2 rangle ) and unit direction

$$leftlangle frac{1}{sqrt{5}}, frac{2}{sqrt{5}} rightrangle = leftlangle frac{sqrt{5}}{5}, frac{2sqrt{5}}{5} rightrangle$$

Higher dimensions

The same formula works in any number of dimensions. ( langle 1, 1, 1, 1 rangle ) has length ( sqrt{4} = 2 ), so its unit vector is ( leftlangle frac12, frac12, frac12, frac12 rightrangle ). In data science, normalizing long vectors like this lets you compare directions (patterns) regardless of size.

Checking your answer

A correct unit vector always passes two quick tests: its components’ squares add to 1, and each component has the same sign as the original vector’s component. For ( langle 2, -1, 2 rangle ), the answer ( leftlangle frac23, -frac13, frac23 rightrangle ) passes both: ( frac49 + frac19 + frac49 = 1 ), and the middle component is still negative.

Related guides

• Pythagorean Theorem
• Distance and Midpoint Formula
• Gradient and Directional Derivative
• Dot Product Calculator

Further reading

• Unit vector (Wikipedia) — normalization, standard basis vectors and direction cosines.

FAQ

What is a unit vector?

A vector with length 1. It shows a direction without any size.

How do you find a unit vector?

Divide the vector by its magnitude: ( hat{u} = frac{v}{|v|} ).

How do you find the magnitude of a vector?

Square each component, add them and take the square root: ( |v| = sqrt{v_1^2 + v_2^2 + v_3^2} ).

What are i, j and k?

The unit vectors along the x-, y- and z-axes: ( mathbf{i} = langle 1, 0, 0 rangle ), ( mathbf{j} = langle 0, 1, 0 rangle ), ( mathbf{k} = langle 0, 0, 1 rangle ).

Is the magnitude ever negative?

No. A length is always zero or positive. Only the zero vector has magnitude 0.

How do I find a vector of a given length in a given direction?

Multiply the unit vector in that direction by the length you want.