Vector Calculator
Add, subtract, scale, dot, cross, normalize, or measure the angle between 2D or 3D vectors. Shows magnitude, the formula, and rounded numeric output.
Vector operations
Formula
Result
(5, 7, 9)
Magnitude: 12.4499
Frequently Asked Questions about the Vector Calculator
What is the difference between dot product and cross product?
Dot product returns a scalar that tells you how aligned two vectors are. Cross product is defined in 3D and returns a vector perpendicular to both inputs, with a length equal to the area of the parallelogram they span.
Why does the calculator return null for a 2D cross product?
In 2D the cross product is a scalar, not a vector, so it does not fit the same output shape as a 3D cross. Use the 3D mode with z = 0 to get the same numeric value, then read the z-component of the result.
How do I find the angle between two vectors?
Use cos(theta) = (a . b) / (|a| * |b|), then take the inverse cosine. The result lies between 0 and 180 degrees. The angle is undefined when either vector has zero length.
What is a unit vector and why is it useful?
A unit vector has length 1 and the same direction as the original. Dividing a vector by its magnitude gives the unit vector. Use it whenever you only care about direction, for example in physics, graphics, or normal calculations.
Can two vectors be added if they have different dimensions?
No. Vector addition requires matching dimensions. To work with a mix, drop missing components (treat a 2D vector as (x, y, 0)) so both vectors live in the same space, then add component by component.