Educational tool; results are estimates for learning, verify with your textbook or instructor.
This tool computes standard Euclidean operations on two vectors $\mathbf{u}$ and $\mathbf{v}$ in $2$ or $3$ dimensions. You can ask for the dot product, a $3$D cross product, a magnitude, or a unit vector in the same direction.
The output is meaningful when the components are coordinates in the usual orthonormal basis, so length and angle have their school meanings. A dot product then equals $|\mathbf{u}||\mathbf{v}|\cos\theta$. A cross product then points along a direction orthogonal to both vectors, with length $|\mathbf{u}||\mathbf{v}|\sin\theta$.
This is not a computer algebra system for abstract vector spaces. It does not handle $n$ components, it does not compute a line integral, and it does not prove Stokes theorem. It is arithmetic on two arrows you typed.
How to use the calculator
Choose $2$D or $3$D. In $2$D the third boxes hide. Enter components for $\mathbf{u}$ and $\mathbf{v}$. The default pair is $\langle 1,0,0\rangle$ and $\langle 0,1,0\rangle$, whose dot product is $0$ and whose cross product is $\langle 0,0,1\rangle$.
Pick an operation and press Compute. Dot product needs both vectors. Cross product needs $3$D. Magnitude and unit vector each use one vector, named in the menu.
How to read the output
A dot product is a single number. Zero means the vectors are orthogonal (or one of them is zero). A large positive value means they point in a similar direction. A negative value means they form an obtuse angle.
A cross product is a vector. It is orthogonal to both inputs when both are nonzero and not parallel. Its direction follows the right hand rule used in most multivariable courses. If you only needed an area of a parallelogram, read the magnitude of that cross product.
A unit vector has length $1$ and the same direction as the source vector. Use it when a later formula wants a direction and not a size.
Common pitfalls
Cross product is not defined as a $3$ vector in $2$D on this page. Switch the dimension menu to $3$D, even if the third components are zero.
The zero vector has no unit vector. The page will say so. Do not invent a direction for a vector with no length.
Components are just numbers. If your data came from a basis that is not orthonormal, the school formulas for angle and length do not apply without a Gram matrix. This tool does not know a custom inner product.
Related reading
Vector language in calculus is mapped on multivariable and vector calculus and on linear algebra. Circulation and curl use these operations: see Stokes theorem intuition and the Stokes theorem, vector, and vector field glossary entries.
Nearby tools: the divergence and curl calculator estimates divergence and curl of a field, the 2 by 2 matrix calculator checks a small matrix, and the 2 by 2 eigenvalue calculator checks stretch factors. A 3 by 3 determinant is the 3 by 3 matrix calculator. Elimination of a system is the reduced row echelon form calculator. Stretch factors of that larger matrix are the 3x3 eigenvalue calculator. A whole list of vectors, turned into an orthogonal basis, is the Gram Schmidt calculator. This page still does not know a custom inner product.
