Inputs
Diagram (2D)
Results
How it's calculated
When to use this vector calculator
Use this page when you need one workspace for vector arithmetic, dot/cross checks, projection, and basis verification. It works well for classroom examples because the diagram, steps, and exports stay in sync with the same inputs.
Start with the right dimension
Choose 2D first for angle, projection, area, and orientation problems on a plane. Switch to 3D when you need a full cross product, triple product, or Gram-Schmidt basis vectors with a non-zero z component.
Check dot, cross, and basis in order
Start with norms and the dot product to confirm lengths and angle logic. Then use the cross product for perpendicular direction or signed-area checks, and finish with Gram-Schmidt when you need an orthogonal classroom verification or a clean basis for later work.
Next steps
FAQ
What operations are supported?
Norms, dot, cross (2D as z), projection and orthogonal decomposition, angle, parallelogram/triangle areas, triple product (volume), and Gram–Schmidt.
Can I share or export results?
Yes. Copy a shareable URL with inputs, export LaTeX for formulas, and download CSV of key values.
Should I start in 2D or 3D?
Start in 2D when you are checking classroom geometry, projection, or area on a plane. Switch to 3D when you need cross products, triple products, or basis vectors that depend on the z component.
When does the cross product need 3D input?
A true vector cross product uses 3D components. In 2D mode this tool embeds vectors on the xy plane and reports the z-direction result, which is enough for signed area checks and orientation tests.
How can I verify projection or basis results in class?
Compare the dot product identities, check whether orthogonal components sum back to the original vector, and confirm basis vectors stay independent after Gram-Schmidt. The step log and diagram make those checks easier to show on a worksheet or whiteboard.
Teacher Notes
Classroom and worksheet notes
For vector arithmetic practice, keep one problem per dimension so angle and basis checks do not mix 2D and 3D assumptions. For dot and cross product lessons, compare the algebraic steps here with a hand-drawn sketch so students can see why orientation or orthogonality changes the result.
If the next question is about coordinate transforms or eigenvectors, move to the eigen page. If the next question is about solving component equations or checking geometric lengths, move to simultaneous equations or the Pythagorean theorem page.