Vector Calculator
Enter two vectors in 2D or 3D and get their sum and difference, magnitudes, unit vectors, dot and cross products, the angle between them and each projection — with exact square roots instead of rounded decimals and the working beside every answer. Switch modes to convert components into magnitude and direction, or back. Free, no sign-up.
Whole numbers, decimals or fractions such as 3/4.
Results
−5
Dot product u · v
5
Magnitude |u|
3
Magnitude |v|
109.4712° ≈ 1.910633 rad
Angle between u and v
| Sum u + v | (3 + 1, 4 + (−2), 0 + 2)(4, 2, 2) |
|---|---|
| Difference u − v | (3 − 1, 4 − (−2), 0 − 2)(2, 6, −2) |
| Scalar multiple 2 × u | (2 × 3, 2 × 4, 2 × 0)(6, 8, 0) |
| Scalar multiple 2 × v | (2 × 1, 2 × (−2), 2 × 2)(2, −4, 4) |
| Magnitude |u| | |u| = √(3² + 4² + 0²) = √255 |
| Unit vector in the direction of u | u ÷ |u|: divide each component by the magnitude(3/5, 4/5, 0) |
| Magnitude |v| | |v| = √(1² + (−2)² + 2²) = √93 |
| Unit vector in the direction of v | v ÷ |v|: divide each component by the magnitude(1/3, −2/3, 2/3) |
| Dot product u · v | u · v = 3 × 1 + 4 × (−2) + 0 × 2−5 |
| Cross product u × v | u × v = (4 × 2 − 0 × (−2), 0 × 1 − 3 × 2, 3 × (−2) − 4 × 1)(8, −6, −10) |
| Area of the parallelogram on u and v | |u × v| = √(|u|²|v|² − (u · v)²); the triangle on u and v has half this area10√2 ≈ 14.142136 |
| Angle between u and v | cos θ = (u · v) / (|u| |v|) = −5 / (5 × 3) = −1/3109.4712° ≈ 1.910633 rad |
| Projection of u onto v | proj = ((u · v) / (v · v)) × v, with factor −5/9(−5/9, 10/9, −10/9) |
| Scalar projection of u along v | comp = (u · v) / |v|, the signed length of the shadow of u along v−5/3 |
| Projection of v onto u | proj = ((v · u) / (u · u)) × u, with factor −1/5(−3/5, −4/5, 0) |
| Scalar projection of v along u | comp = (u · v) / |v|, the signed length of the shadow of v along u−1 |
Three-dimensional vectors are not drawn; the exact results above are complete.
How to Use the Vector Calculator
- 1. Choose a mode and a dimension. "Work with two vectors" does every operation at once; the other two modes convert between components and magnitude-and-direction.
- 2. Enter the components. Whole numbers, decimals and fractions are all exact. Pick 2D or 3D first, and fill in u and v (and a scalar k if you want multiples).
- 3. Read the results. Each row gives its formula with your numbers substituted, then the answer. Magnitudes stay as radicals — |(1, 1)| is √2 — with a decimal beside them.
- 4. Check the diagram. In 2D, u, v and u + v are drawn to scale from the origin with the parallelogram completed and the foot of the projection marked.
The Formulas
| Quantity | Formula | Notes |
|---|---|---|
| Magnitude | |v| = √(x² + y² + z²) | The length of the vector |
| Unit vector | v̂ = v / |v| | Length 1, same direction; undefined for the zero vector |
| Dot product | u · v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v| cos θ | A number; 0 means perpendicular |
| Cross product | u × v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁) | 3D only; perpendicular to both; |u × v| = |u||v| sin θ |
| Angle | θ = cos⁻¹((u · v) / (|u||v|)) | Between 0° and 180° |
| Vector projection | proj_v u = ((u · v) / (v · v)) v | The part of u along v |
| Scalar projection | comp_v u = (u · v) / |v| | Signed length of that part |
For 2D vectors the calculator treats them as lying in the xy-plane, so the cross product reduces to the single number u₁v₂ − u₂v₁ — the z-component.
Worked Example: u = (3, 4, 0) and v = (1, −2, 2)
- Sum and difference. u + v = (4, 2, 2) and u − v = (2, 6, −2).
- Magnitudes. |u| = √(9 + 16 + 0) = 5 and |v| = √(1 + 4 + 4) = 3, so the unit vectors are (3/5, 4/5, 0) and (1/3, −2/3, 2/3).
- Dot product. u · v = 3 − 8 + 0 = −5. It is negative, so the angle is obtuse.
- Angle. cos θ = −5 / (5 × 3) = −1/3, so θ ≈ 109.4712° (about 1.910633 rad).
- Cross product. u × v = (8, −6, −10). Its dot product with u is 24 − 24 + 0 = 0 and with v it is 8 + 12 − 20 = 0, so it really is perpendicular to both.
- Parallelogram area. |u × v| = √(64 + 36 + 100) = √200 = 10√2 ≈ 14.142136.
- Projection of u onto v. The factor is −5/9, so the projection is (−5/9, 10/9, −10/9); the scalar projection is −5/3.
Magnitude and Direction
In 2D, a vector with components (x, y) has magnitude √(x² + y²) and direction angle θ measured counterclockwise from the positive x-axis. The calculator uses atan2(y, x), which puts θ in the right quadrant — the plain arctan(y/x) cannot tell (−3, 4) from (3, −4). So (−3, 4) has magnitude 5 and θ ≈ 126.8699°, in quadrant II. Going the other way, x = r cos θ and y = r sin θ: a vector of length 10 at 30° has components (5√3, 5), which the tool keeps exact for multiples of 30° and 45°.
In 3D a vector's direction is given by the angles α, β and γ it makes with the x-, y- and z-axes. Their cosines, v_x/|v|, v_y/|v| and v_z/|v|, are the direction cosines, and their squares always sum to 1. For (1, 2, 2) they are 1/3, 2/3 and 2/3, giving angles of about 70.5288°, 48.1897° and 48.1897°.
Frequently Asked Questions
- What is the difference between the dot product and the cross product?
- The dot product of two vectors is a single number, |u||v| cos θ, that measures how far they point the same way. The cross product is a vector perpendicular to both, with length |u||v| sin θ equal to the area of the parallelogram they span. The cross product exists in 3D; the dot product works in any dimension.
- How do you find the angle between two vectors?
- Divide the dot product by the product of the magnitudes to get cos θ, then take the inverse cosine: θ = cos⁻¹((u · v) / (|u||v|)). The result is always between 0° and 180°. A dot product of 0 gives 90°, and the angle is undefined if either vector is the zero vector.
- How do you find a unit vector?
- Divide the vector by its own magnitude. The result points the same way and has length exactly 1. For (1, 1) that is (1/√2, 1/√2), which the calculator writes in rationalized form as (√2/2, √2/2).
- How do you know if two vectors are perpendicular or parallel?
- They are perpendicular exactly when their dot product is 0. They are parallel when one is a scalar multiple of the other, which is exactly when their cross product is the zero vector (in 2D, when u₁v₂ − u₂v₁ = 0).
- What does the projection of one vector onto another mean?
- The projection of u onto v is the component of u that lies along v: the shadow u casts on the line through v. The scalar projection is that shadow's signed length, negative if u points against v. What is left, u minus the projection, is perpendicular to v.
- Why does the calculator show square roots instead of decimals?
- Exact radicals are what a textbook or an exam expects, and they avoid rounding error. Each radical comes with a six-decimal approximation next to it, so you can use whichever form you need.