Calculate vector addition, subtraction, dot and cross products, magnitudes, angles, distances, unit vectors and projections with instant results.
For 2D vectors, enter 0 for each z component
A vector calculator performs mathematical operations on quantities that have both magnitude and direction. Vectors are commonly represented by x, y and z coordinate components.
This tool supports two-dimensional and three-dimensional calculations. For a 2D vector, simply enter zero as the z component.
You can calculate addition, subtraction, dot products, cross products, vector lengths, angles, distances, unit vectors and projections from one calculator.
Complete several vector calculations in three simple steps.
Add the x, y and z components of the first vector. Enter zero for z when using a 2D vector.
Add the coordinate components of the second vector using positive, negative or decimal values.
Calculate vector operations, measurements, directions and projections instantly.
These formulas are used to calculate relationships between two-dimensional and three-dimensional vectors.
Add or subtract corresponding vector components.
Multiply matching components and add the products.
The cross product produces a vector perpendicular to both original vectors.
The angle is calculated using the dot product and vector magnitudes.
A unit vector has the same direction as the original vector but a magnitude of one.
Projection measures the portion of Vector A acting in the direction of Vector B.
Review common vector inputs and their calculated results.
| Vector A | Vector B | A + B | A · B | A × B |
|---|---|---|---|---|
| ⟨1, 2, 3⟩ | ⟨4, 5, 6⟩ | ⟨5, 7, 9⟩ | 32 | ⟨−3, 6, −3⟩ |
| ⟨3, 4, 0⟩ | ⟨2, −1, 5⟩ | ⟨5, 3, 5⟩ | 2 | ⟨20, −15, −11⟩ |
| ⟨2, 0, 0⟩ | ⟨0, 3, 0⟩ | ⟨2, 3, 0⟩ | 0 | ⟨0, 0, 6⟩ |
| ⟨−2, 5, 1⟩ | ⟨3, 0, −4⟩ | ⟨1, 5, −3⟩ | −10 | ⟨−20, −5, −15⟩ |
Find answers to common questions about vectors, products, magnitudes and projections.
A vector is a mathematical quantity with both magnitude and direction. It may be represented using coordinate components such as ⟨x, y⟩ or ⟨x, y, z⟩.
Yes. Enter zero for the z component of both vectors to perform calculations using two-dimensional vectors.
Square each component, add the squared values and take the square root. For ⟨x, y, z⟩, the magnitude is √(x² + y² + z²).
The dot product is a scalar value that indicates how strongly two vectors point in the same direction. A dot product of zero means nonzero vectors are perpendicular.
The cross product produces a vector perpendicular to both original vectors. It is commonly used in geometry, physics and three-dimensional graphics.
The angle is undefined when either input is the zero vector because a zero vector has no defined direction.
A unit vector has a magnitude of exactly one while retaining the direction of its original vector.
Vector projection determines how much of one vector acts in the direction of another vector. Projection onto the zero vector is undefined.
Enter two vectors above to calculate their operations, lengths, angle, unit directions and projections.